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Archive-name: puzzles/archive/arithmetic/part1
Last-modified: 17 Aug 1993
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```==> arithmetic/7-11.p <==
A customer at a 7-11 store selected four items to buy, and was told
that the cost was \$7.11.  He was curious that the cost was the same
as the store name, so he inquired as to how the figure was derived.
The clerk said that he had simply multiplied the prices of the four
individual  items.  The customer  protested  that  the four  prices
should have been ADDED,  not MULTIPLIED.  The  clerk said that that
was OK with him, but, the result was still the same: exactly \$7.11.

What were the prices of the four items?

==> arithmetic/7-11.s <==
The prices are: \$1.20, \$1.25, \$1.50, and \$3.16

\$7.11 is not the only number which works.  Here are the first 160 such
numbers, preceded by a count of distinct solutions for that price.
Note that \$7.11 has a single, unique solution.

1 -  \$6.44      1 -  \$7.83      2 -  \$9.20      3 - \$10.89
1 -  \$6.51      1 -  \$7.86      1 -  \$9.23      1 - \$10.95
1 -  \$6.60      3 -  \$7.92      1 -  \$9.24      2 - \$11.00
1 -  \$6.63      1 -  \$8.00      1 -  \$9.27      1 - \$11.07
1 -  \$6.65      1 -  \$8.01      2 -  \$9.35      1 - \$11.13
1 -  \$6.72      1 -  \$8.03      3 -  \$9.36      1 - \$11.16
2 -  \$6.75      5 -  \$8.10      1 -  \$9.38      1 - \$11.22
1 -  \$6.78      1 -  \$8.12      5 -  \$9.45      2 - \$11.25
1 -  \$6.80      1 -  \$8.16      2 -  \$9.48      2 - \$11.27
2 -  \$6.84      2 -  \$8.19      1 -  \$9.54      1 - \$11.30
1 -  \$6.86      1 -  \$8.22      1 -  \$9.57      1 - \$11.36
1 -  \$6.89      1 -  \$8.25      1 -  \$9.59      1 - \$11.40
2 -  \$6.93      3 -  \$8.28      2 -  \$9.60      2 - \$11.43
1 -  \$7.02      3 -  \$8.33      1 -  \$9.62      2 - \$11.52
1 -  \$7.05      1 -  \$8.36      2 -  \$9.63      2 - \$11.55
2 -  \$7.07      1 -  \$8.37      1 -  \$9.66      2 - \$11.61
1 -  \$7.08      2 -  \$8.40      1 -  \$9.68      1 - \$11.69
1 -  \$7.11      1 -  \$8.45      2 -  \$9.69      1 - \$11.70
1 -  \$7.13      2 -  \$8.46      1 -  \$9.78      1 - \$11.88
2 -  \$7.14      1 -  \$8.52      2 -  \$9.80      1 - \$11.90
3 -  \$7.20      5 -  \$8.55      1 -  \$9.81      1 - \$11.99
1 -  \$7.25      1 -  \$8.60      1 -  \$9.87      1 - \$12.06
1 -  \$7.26      4 -  \$8.64      4 -  \$9.90      1 - \$12.15
2 -  \$7.28      1 -  \$8.67      1 -  \$9.92      1 - \$12.18
1 -  \$7.29      1 -  \$8.69      2 -  \$9.99      1 - \$12.24
3 -  \$7.35      1 -  \$8.73      1 - \$10.01      1 - \$12.30
1 -  \$7.37      2 -  \$8.75      1 - \$10.05      1 - \$12.32
1 -  \$7.47      1 -  \$8.76      2 - \$10.08      1 - \$12.35
1 -  \$7.50      1 -  \$8.78      1 - \$10.17      2 - \$12.42
1 -  \$7.52      5 -  \$8.82      1 - \$10.20      1 - \$12.51
4 -  \$7.56      1 -  \$8.85      2 - \$10.26      1 - \$12.65
1 -  \$7.62      1 -  \$8.88      3 - \$10.29      2 - \$12.69
4 -  \$7.65      2 -  \$8.91      3 - \$10.35      1 - \$12.75
1 -  \$7.67      1 -  \$8.94      2 - \$10.44      1 - \$12.92
2 -  \$7.70      1 -  \$8.96      1 - \$10.53      1 - \$12.96
3 -  \$7.74      3 -  \$9.00      1 - \$10.56      1 - \$13.23
1 -  \$7.77      1 -  \$9.02      1 - \$10.64      1 - \$13.41
1 -  \$7.79      2 -  \$9.03      2 - \$10.71      1 - \$13.56
2 -  \$7.80      1 -  \$9.12      3 - \$10.80      1 - \$14.49
1 -  \$7.82      2 -  \$9.18      1 - \$10.85      1 - \$15.18

There are plenty of solutions for five summands.  Here are a few:

\$8.28  -- at least two solutions
\$8.47  -- at least two solutions
\$8.82  -- at least two solutions
--
Mark Johnson       mark@microunity.com       (408) 734-8100

There may be many approximate solutions, for example: \$1.01, \$1.15, \$2.41,
and \$2.54.  These sum to \$7.11 but the product is 7.1100061.

==> arithmetic/arithmetic.progression.p <==
Is there an arithmetic progression of 20 or more primes?

==> arithmetic/arithmetic.progression.s <==
There is an arithmetic progression of 21 primes:
142072321123 + 1419763024680 i, 0 <= i < 21.

It was discovered on 30 November 1990, by programs running in the background
on a network of Sun 3 workstations in the Department of Computer Science,
University of Queensland, Australia.

See: Andrew Moran and Paul Pritchard, The design of a background job
on a local area network, Procs. 14th Australian Computer Science Conference,
1991, to appear.

==> arithmetic/clock/day.of.week.p <==
It's restful sitting in Tom's cosy den, talking quietly and sipping
a glass of his Madeira.

I was there one Sunday and we had the usual business of his clock.
When the radio gave the time at the hour, the Ormolu antique was
exactly 3 minutes slow.

"It loses 7 minutes every hour", my old friend told me, as he had done
so many times before.  "No more and no less, but I've gotten used to
it that way."

When I spent a second evening with him later that same month, I remarked
on the fact that the clock was dead right by radio time at the hour.
It was rather late in the evening, but Tom assured me that his treasure
had not been adjusted nor fixed since my last visit.

What day of the week was the second visit?

From "Mathematical Diversions" by Hunter + Madachy

==> arithmetic/clock/day.of.week.s <==
The answer is 17 days and 3 hours later, which would have been a Wednesday.
This is the only other time in the same month when the two would agree at all.

In 17 days the slow clock loses 17*24*7 minutes = 2856 minutes,
or 47 hours and 36 minutes.  In 3 hours more it loses 21 minutes, so
it has lost a total of 47 hours and 57 minutes.  Modulo 12 hours, it
has *gained* 3 minutes so as to make up the 3 minutes it was slow on
Sunday.  It is now (fortnight plus 3 days) exactly accurate.

Since the clock was not adjusted since the last visit, it's also
possible that the radio time shifted by one hour due to a change to or
from summer daylight saving time.  However, it turns out that the only
additional possibilities that need to be considered are those of 4 days
15 hours later, when the clock would have lost 12 hours 57 minutes, and
29 days 15 hours later, when the clock would have lost 3 days 10 hours
57 minutes.  Without even considering the rules for when in the month the
clock is changed, these possible solutions are ruled out because we know
that both visits were in the evening ("I spent a second evening with him").
and they involve times in a different part of the day.

==> arithmetic/clock/palindromic.p <==
How many times per day does a digital clock display a palindromic number?

==> arithmetic/clock/palindromic.s <==
The problem is underspecified.  Digital clocks may run from

(a) 1:00 to 12:59
(b) 01:00 to 12:59
(c) 0:00 to 23:59
(d) 00:00 to 23:59
(e-h) any of the above with seconds appended, :00 to :59.

I agree that not all of these are common, but I have seen some uncommon
ones.  For that matter, I've seen ones not on the above list -- the Toronto
subway stations used to contain digital clocks that ran from 0:00 to 12:59
(pm), then from 1:00 (pm) to 11:59 (pm), while a computer that I used to
use had the inverse anomaly -- its time of day command displayed times
from 12:00:00 to 12:59:59 (am), then 01:00:00 to 23:59:59!

I get the following results for the 8 cases.

CASE	AM+PM	HOURS pals/hr  TOTAL	OVERALL TOTAL
(a)	yes	1-9	 6	 54
10-12	 1	  3	114

(b)	yes	01-05	 1	  5
06-09	 0	  0
10-12	 1	  3	16

(c)	no	0-9	 6	 60
10-15	 1	  6
16-19	 0	  0
20-23	 1	  4	70

(d)	no	00-05	 1	  6
06-09	 0	  0
10-15	 1	  6
16-19	 0	  0
20-23	 1	  4	16

(e)	yes	1-9	60	540
10-12	 6	 18	1116

(f)	yes	01-05	 6	 30
06-09	 0	  0
10-12	 6	 18	96

(g)	no	0-9	60	600
10-15	 6	 36
16-19	 0	  0
20-23	 6	 24	660

(h)	no	00-05	 6	 36
06-09	 0	  0
10-15	 6	 36
16-19	 0	  0
20-23	 6	 24	96

--Mark Brader (msb@sq.com)

==> arithmetic/clock/reversible.p <==
How many times per day can the hour and minute hands on an analog clock switch
roles and still signify a valid time, ignoring the second hand?

==> arithmetic/clock/reversible.s <==
Have 12 clocks C1, C2 ... C12 show 1:00, 2:00, ..., 12:00.  Have a
clock C0 show 12:00

Now turn C0 around 12 hours, simultaneously turning C1-C12 so their
hour hands always coincide with the minute hand of C0, i.e., as C0
spans 12 hours, C1-C12 will span 1 hour, but for each possible placing
of the hour hand, all 12 possible 'true' placings of the minute hand
will be represented by one of the 12 clocks.

Each time the hour hand of C0 coincides with the minute hand of a
C1-C12 clock we have a reversible valid time.  This happens regularly
12 times each C0 hour, but the first and last time is equal (12:00), so
the number of reversible true times is 12*12-1 = 143 spaced regularly
in the 12-hour interval, ie. each 5 min 2.0979+ sec

--
stein.kulseth@tf.tele.no [X.400] stein.kulseth@nta.no [internet]

==> arithmetic/clock/right.angle.p <==
How many times per day do the hour and minute hands of a clock form a
right angle?

==> arithmetic/clock/right.angle.s <==
44.  Twice each hour equals 48, less one between 2:00 and 4:00 and one between
8:00 and 10:00 for both A.M. and P.M.

==> arithmetic/clock/thirds.p <==
Do the 3 hands on a clock ever divide the face of the clock into 3
equal segments, i.e. 120 degrees between each hand?

==> arithmetic/clock/thirds.s <==
First let us assume that our clock has 60 divisions.  We will show that
any time the hour hand and the minute hand are 20 divisions (120 degrees)
apart, the second hand cannot be an integral number of divisions from the
other hands, unless it is straight up (on the minute).

Let us use h for hours, m for minutes, and s for seconds.
We will use =n to mean congruent mod n, thus 12 =5 7.

We know that m =60 12h, that is, the minute hand moves 12 times as fast
as the hour hand, and wraps around at 60.
We also have s =60 60m. This simplifies to s/60 =1 m, which goes to
s/60 = frac(m) (assuming s is in the range 0 <= s < 60), which goes to
s = 60 frac(m).  Thus, if m is 5.5, s is 30.

Now let us assume the minute hand is 20 divisions ahead of the hour hand.
So m =60 h + 20, thus 12h =60 h + 20, 11h =60 20, and, finally,
h =60/11 20/11 (read 'h is congruent mod 60/11 to 20/11').
So all values of m are k + n/11 for some integral k and integral n,
0 <= n < 11.  s is therefore 60n/11.  If s is to be an integral number of
units from m and h, we must have 60n =11 n.  But 60 and 11 are relatively
prime, so this holds only for n = 0.  But if n = 0, m is integral, so
s is 0.

Now assume, instead, that the minute hand is 20 divisions behind the hour hand.
So m =60 h - 20, 12h =60 h - 20, 11h =60 -20, h =60/11 -20/11.
So m is still k + n/11.  Thus s must be 0.

But if s is 0, h must be 20 or 40.  But this translates to 4 o'clock or
8 o'clock, at both of which the minute hand is at 0, along with the second
hand.

Thus the 3 hands can never be 120 degrees apart, Q.E.D.

This assumes, of course, that the second hand is synchronized with the
minute hand.  This is not true on some inexpensive analog watches.  This
also assumes that the watch is not broken (:^)).

==> arithmetic/consecutive.composites.p <==
Are there 10,000 consecutive non-prime numbers?

==> arithmetic/consecutive.composites.s <==
9973!+2 through 9973!+10006 are composite.

==> arithmetic/consecutive.product.p <==
Prove that the product of three or more consecutive positive integers cannot
be a perfect square.

==> arithmetic/consecutive.product.s <==
Three consecutive numbers:
If a and b are relatively prime, and ab is a square,
then a and b are squares. (This is left as an exercise.)

Suppose (n - 1)n(n + 1) = k^2, where n > 1.
Then n(n^2 - 1) = k^2.  But n and (n^2 - 1) are relatively prime.
Therefore n^2 - 1 is a perfect square, which is a contradiction.

Four consecutive numbers:
n(n + 1)(n + 2)(n + 3) = (n^2 + 3n + 1)^2 - 1

Five consecutive numbers:
Assume the product is a integer square, call it m.

The prime factorization of m must have even numbers of each prime factor.

For each prime factor, p, of m, p >= 5, p^2k must divide one of the
consecutive naturals in the product.  (Otherwise, the difference between two
of the naturals in the product would be a positive multiple of a prime >= 5.
But in this problem, the greatest difference is 4.) So we need only consider
the primes 2 and 3.

Each of the consecutive naturals is one of:
1)	a perfect square
2)	2 times a perfect square
3)	3 times a perfect square
4)	6 times a perfect square.

By the shoe box principle, two of the five consecutive numbers must fall into
the same category.

If there are two perfect squares, then their difference being less than five
limits their values to be 1 and 4.  (0 is not a natural number, so 0 and 1
and 0 and 4 cannot be the perfect squares.)  But 1*2*3*4*5=120!=x*x where x
is an integer.

If there are two numbers that are 2 times a perfect square, then their
difference being less than five implies that the perfect squares (which are
multiplied by 2) are less than 3 apart, and no two natural squares differ by
only 1 or 2.

A similar argument holds for two numbers which are 3 times a perfect square.

We cannot have the case that two of the 5 consecutive numbers are multiples
(much less square multiples) of 6, since their difference would be >= 6, and
our span of five consecutive numbers is only 4.

Therefore the assumption that m is a perfect square does not hold.

QED.

In general the equation:

y^2 = x(x+1)(x+2)...(x+n),    n > 3

has only the solution corresponding to y = 0.

This is a theorem of Rigge [O. Rigge, ``Uber ein diophantisches Problem'',
IX Skan. Math. Kong. Helsingfors (1938)] and Erdos [P. Erdos, ``Note on
products of consecutive integers,'' J. London Math. Soc. #14 (1939),
pages 194-198].

A proof can be found on page 276 of [L. Mordell, ``Diophantine
Equations'', Academic Press 1969].

==> arithmetic/consecutive.sums.p <==
Find all series of consecutive positive integers whose sum is exactly 10,000.

==> arithmetic/consecutive.sums.s <==
Generalize to find X (and I) such that
(X + X+1 + X+2 + ... + X+I) = T
for any integer T.

You are asking for all (X,I) s.t. (2X+I)(I+1) = 2T.  The problem is
(very) slightly easier if we don't restrict X to being positive, so
we'll solve this first.

Note that 2X+I and I+1 must have different parities, so the answer
to the relaxed question is N = 2*(o_1+1)*(o_2+1)*...*(o_n+1), where
2T = 2^o_0*3^o_1*...*p_n^o_n (the prime factorization); this is easily
seen to be the number of ways we can break 2T up into two positive
factors of differing parity (with order).

In particular, 20000 = 2^5*5^4, hence there are 2*(4+1) = 10 solutions
for T = 10000.  These are (2X+I,I+1):

(32*1,5^4)   (32*5,5^3)   (32*5^2,5^2)   (32*5^3,5)   (32*5^4,1)
(5^4,32*1)   (5^3,32*5)   (5^2,32*5^2)   (5,32*5^3)   (1,32*5^4)

And they give rise to the solutions (X,I):

(-296,624)   (28,124)   (388,24)   (1998,4)     (10000,0)
(297,31)     (-27,179)  (-387,799) (-1997,3999) (-9999,19999)

If you require that X>0 note that this is true iff 2X+I > I+1 and
hence the number of solutions to this problem is N/2 (due to the
symmetry of the above ordered pairs).

==> arithmetic/conway.p <==
Describe the sequence a(1)=a(2)=1, a(n) = a(a(n-1)) + a(n-a(n-1)) for n>2.

==> arithmetic/conway.s <==
This sequence and its remarkable properties were discovered, apparently
independently, by Douglas Hofstadter (circa 1975), John Conway (in the
early 1980's), B.W. Connoly, and others. Since Conway discovered many of
the deeper properties, and is the one responsible for popularizing the
sequence, it seems appropriate to name the sequence after him.

Some properties of a(n):

--  a(n+1) - a(n) = 0 or 1

--  a(2^n) = 2^(n-1)

--  n/2 <= a(n) <= n

--  A(n)= 2a(n) - n    has zeros at the powers of 2 and
positive "humps" in between

--  A(2^n + t) = A(2^(n+1) - t)  for t between 0 and 2^n,
i.e. the humps are symmetric

-- a(2n) <= 2a(n)

-- a(n)/n --> 1/2  as  n --> infinity

-- a(n) and A(n) are self-similar, in the sense that their values on the
(N+1)'st hump can be obtained by a "magnification" process applied
to the values on the N'th hump. They are *not* chaotic sequences,
since their asymptotics and combinatorial structure are very regular
and rigid.

In a lecture at Bell Labs in 1988, Conway asked for the rate at
which  a(n)/n converges to 1/2.  In particular, he asked for
N(epsilon), the last value of  n  such that  |a(n)/n - 1/2|
exceeds epsilon, in the case  epsilon=1/20.  This problem was
solved by Colin Mallows of Bell Labs: he found that  N(1/20)=1489
and N(1/40)=6083008742. These values are reported in his article
in the American Mathematical Monthly, January 1991, p. 5.

Conway's sequence has a deep combinatorial structure, and is intimately
connected with all of the following: variants of Pascal's triangle, the
Gaussian distribution, combinatorial operations on finite sets,
coin-pushing games, and Fibonacci and Catalan numbers.  All of this is
explained in my joint paper with Ravi Vakil; anyone who would like to
receive a copy in LaTex format should contact me at the e-mail address
listed below.

One byproduct of our work is an algorithm to determine  N(epsilon) for
given positive epsilon.  Here are some particular values:

e      Last  n  such that   |a(n)/n - 1/2| > e
----    ----------------------------------------------------------
1/20    1489  (found by Mallows in 1988)
1/30    758765
1/40    6083008742  (found by Mallows in 1988)
1/50    809308036481621
1/60    1684539346496977501739
1/70    55738373698123373661810220400
1/80    15088841875190938484828948428612052839
1/90    127565909103887972767169084026274554426122918035
1/100   8826608001127077619581589939550531021943059906967127007025

These values were computed by the Mathematica program listed below; the
algorithm is explained in our paper.  To use the program, load it into
Mathematica and type   neps[t]  for some numerical value of  t  (which
should probably be smaller than 1/5 or so). The program will output  N(t),
e.g.  neps[1/20] = 1489. These numbers grow very fast: N(epsilon) will be
about 2^((0.138015/epsilon)^2), so  N(1/1000) will have about 5735 digits.
The program requires very little memory space, but its runtime appears to
grow rapidly as epsilon decreases (on a Sun 3, it took about one second to
find the first value listed, and several minutes to find the last).

neps[eps_] := Block[{W}, L = 1 + Floor[(0.138015/eps)^2]; e = eps*2;
W = processvector[L]; While[Length[W] > 0,
nmax = 1 + Last[W][]; L++; W = processvector[L]]; nmax]

processvector[L_] :=
Block[{V}, V = startvector[L]; While[notfound[V], V = newbody[V]]; V]

startvector[L_] := Select[initialvector[L], gt]

initialvector[L_] :=
Table[{L, b, Binomial[L - 1, b - 1],
2^(L + 1) - Sum[Binomial[L, c], {c, b, L}]}, {b, 0, L - 1}]

c = 0

c[n_] := c[n] = Sum[Binomial[2*a, a]/(a + 1), {a, 0, n - 1}]

gt[x_] := U[x] > e

U[x_] := (x[] + M[x[], x[]])/(x[] + incp[x[], x[]])

M[n_, n_] = -1

M[n_, k_] := Binomial[n - 1, K[n, k]] - Binomial[n - 1, k - 1] + c[K[n, k]]

K[n_, k_] := Min[k, n - k]

incp[n_, k_] := Max[M[n, k], 1]

notfound[V_] :=
Length[V] > 0 && Last[V][] > 0 && Last[V][] > Last[V][]

newbody[V_] := Join[Drop[V, -1], newtail[V]]

newtail[V_] := Select[{vleft[Last[V]], vright[Last[V]]}, gt]

vleft[x_] := {x[] - 1, x[] - 1, x[], x[]}

vright[x_] := {x[] - 1, x[], x[] + S[x[] - 1, x[] - 1],
x[] + Binomial[x[] - 1, x[] - 1]}

S[n_, k_] := Binomial[n - 1, k] - Binomial[n - 1, k - 1]

-Tal Kubo     kubo@math.harvard.edu  or   kubo@zariski.harvard.edu

==> arithmetic/digits/6.and.7.p <==
Does every number which is not divisible by 5 have a multiple whose
only digits are 6 and 7?

==> arithmetic/digits/6.and.7.s <==
Yes.  My proof follows:

Claim 1: For every k, there is a k-digit number whose only digits
are 6 and 7, which is divisible by 2^k.

The proof is by induction.  Suppose N is a k-digit number
satisfying the above condition.  Then either N = 0 (mod 2^(k+1))
or N = 2^k (mod 2^(k+1)). Note that 6(10^k) = 0 (mod 2^(k+1)),
and 7(10^k) = 2^k (mod 2^(k+1)).  So, either 6*10^k + N or
7*10^k + N is divisible by 2^(k+1).

Claim 2:  If m and 10 are relatively prime, then for any r,
there is a number N whose only digits are 6 and 7 such that
N = r (mod m).

Proof:  Let K be the (m^2)-digit number whose only digit is 6.
There is an s, 0 <= s < m, so that K + s = r (mod m).
Let N = K + 10^(m - 1) + 10^(2m - 2) + . . . + 10^(sm - s).
Since 10^(im - i) = 1 (mod m), N = K + s (mod m) = r (mod m).
Clearly, every digit of N is either 6 or 7.

Claim 3: If n is not divisible by 5, then there is a number N whose
only digits are 6 and 7, so that N is divisible by n.

Proof:  We can write n = (2^k)m, with gcd(m,10)=1.
Use claim 1 to find a k-digit number M, whose only digits are 6 and 7,
which is divisible by 2^k.  Choose an integer r so that
(10^k)r + M = 0 (mod m).  Use claim 2 to find a number K whose
only digits are 6 and 7, so that K = r (mod m).  Let N = 10^k K + M.
Then N = 0 (mod m) and N = 0 (mod 2^k), so N is divisible by n.
Finally, the only digits of N are 6 and 7, so we are done.

--
David Radcliffe                                radcliff@csd4.csd.uwm.edu

==> arithmetic/digits/all.ones.p <==
Prove that some multiple of any integer ending in 3 contains all 1s.

==> arithmetic/digits/all.ones.s <==
Let n be our integer; one such desired multiple is then
(10^(phi(n))-1)/9.  All we need is that (n,10) = 1, and if the last
digit is 3 this must be the case.  A different proof using the
pigeonhole principle is to consider the sequence 1, 11, 111, ..., (10^n
- 1)/9.  We must have at some point that either some member of our
sequence = 0 (mod n) or else some value (mod n) is duplicated.  Assume
the latter, with x_a and x_b, x_b>x_a,  possesing the duplicated
remainders.  We then have that x_b - x-a = 0 (mod n).  Let m be the
highest power of 10 dividing x_b - x_a.  Now since (10,n) = 1, we can
divide by 10^m and get that (x_b - x_a)/10^m = 0 (n).  But (x_b -
x_a)/10^m is a number containing only the digit 1.

Q.E.D.

==> arithmetic/digits/arabian.p <==
What is the Arabian Nights factorial, the number x such that x! has 1001
digits?  How about the prime x such that x! has exactly 1001 zeroes on
the tail end.  (Bonus question, what is the 'rightmost' non-zero digit in x!?)

==> arithmetic/digits/arabian.s <==
The first answer is 450!.

Determining the number of zeroes at the end of x! is relatively easy once
you realize that each such zero comes from a factor of 10 in the product

1 * 2 * 3 * ... * x

Each factor of 10, in turn, comes from a factor of 5 and a factor of 2.
Since there are many more factors of 2 than factors of 5, the number of 5s
determines the number of zeroes at the end of the factorial.

The number of 5s in the set of numbers 1 .. x (and therefore the number
of zeroes at the end of x!) is:

z(x) = int(x/5) + int(x/25) + int(x/125) + int(x/625) + ...

This series terminates when the powers of 5 exceed x.

I know of no simple way to invert the above formula (i.e., to find x for
a given z(x)), but I can approximate it by noting that, except for the "int"
function,

5*z(x) - x = z(x)

which gives:

x = 4*z(x) (approximately).

The given problem asked, "For what prime x is z(x)=1001".  By the above forumla,
this is approximately 4*1001=4004.  However, 4004! has only

800 + 160 + 32 + 6 + 1 = 999 zeroes at the end of it.

The numbers 4005! through 4009! all have 1000 zeroes at their end and
the numbers 4010! through 4014! all have 1001 zeroes at their end.

Since the problem asked for a prime x, and 4011 = 3*7*191, the only solution
is x=4013.

The problem of determining the rightmost nonzero digit in x! is somewhat more
difficult.  If we took the numbers 1, 2, ... , x and removed all factors of 5
(and an equal number of factors of 2), the remaining numbers multiplied
together modulo 10 would be the answer.  Note that since there are still many
factors of 2 left, the rightmost nonzero digit must be 2, 4, 6, or 8 (x > 1).

Letting r(x) be the rightmost nonzero digit in x!, an expression for r(x) is:

r(x) = (r(int(x/5)) * w * r(x mod 10)) mod 10, x >= 10.

where w is 4 if int(x/10) is odd and 6 if it is even.

The values of r(x) for 0 <= x <= 9 are 1, 1, 2, 6, 4, 2, 2, 4, 2, and 8.

The way to see this is true is to take the numbers 1, 2, ..., x in groups
of 10.  In each group, remove 2 factors of 10.  For example, from the
set 1, 2, ..., 10, choose a factor of 2 from 2 and 6 and a factor of 5 from
5 and 10.  This leaves 1, 1, 3, 4, 1, 3, 7, 8, 9, 2.  Next, separate all the
factors that came from multiples of 5.  The rightmost nonzero digit of x!
can now (hopefully) be seen to be:

r(x) = (r(int(x/5)) * w * r(x mod 10)) mod 10

where w is the rightmost digit in the number formed by multiplying the numbers
1, 2, 3, ..., 10*int(x/10) after the factors of 10 and the factors left over
by the multiples of 5 have been removed.  In the example with x = 10, this
would be (1 * 1 * 3 * 4 * 3 * 7 * 8 * 9) mod 10 = 4.  The "r(x mod 10)" term
takes care of the numbers from 10*int(x/10)+1 up to x.

The "w" term can be seen to be 4 or 6 depending on whether int(x/10) is odd or
even since, after removing 10*n+5 and 10*n+10 and a factor of 2 each from
10*n+2 and 10*n+6 from the group of numbers 10*n+1 through 10*n+10, the
remaining factors (mod 10) always equals 4 and 4^t mod 10 = 4 if t is odd and
6 when t is even (t != 0).

So, finally, the rightmost nonzero digit in 4013! is found as follows:

r(4013) = (r(802) * 4 * 6) mod 10
r(802)  = (r(160) * 6 * 2) mod 10
r(160)  = (r(32)  * 6 * 1) mod 10
r(32)   = (r(6)   * 4 * 2) mod 10

Using a table of r(x) for 0 <= x <= 9, r(6) = 2.  Then,

r(32)   = (2 * 4 * 2) mod 10 = 6
r(160)  = (6 * 6 * 1) mod 10 = 6
r(802)  = (6 * 6 * 2) mod 10 = 2
r(4013) = (2 * 4 * 6) mod 10 = 8

Thus, the rightmost nonzero digit in 4013! is 8.

==> arithmetic/digits/circular.p <==
What 6 digit number, with 6 different digits, when multiplied by all integers
up to 6, circulates its digits through all 6 possible positions, as follows:
ABCDEF * 1 = ABCDEF
ABCDEF * 3 = BCDEFA
ABCDEF * 2 = CDEFAB
ABCDEF * 6 = DEFABC
ABCDEF * 4 = EFABCD
ABCDEF * 5 = FABCDE

==> arithmetic/digits/circular.s <==
ABCDEF=142857 (the digits of the expansion of 1/7).

==> arithmetic/digits/divisible.p <==
Find the least number using 0-9 exactly once that is evenly divisible by each
of these digits.

==> arithmetic/digits/divisible.s <==
Since the sum of the digits is 45, any permutation of the digits gives a
multiple of 9.  To get a multiple of both 2 and 5, the last digit must
be 0, and thus to get a multiple of 8 (and 4), the tens digit must be
even, and the hundreds digit must be odd if the tens digit is 2 or 6,
and even otherwise.  The number will also be divisible by 6, since it is
divisible by 2 and 3, so 7 is all we need to check.  First, we will look
for a number whose first five digits are 12345; now, 1234500000 has a
remainder of 6 when divided by 7, so we have to arrange the remaining
digits to get a remainder of 1.  The possible arrangements, in
increasing order, are

78960, remainder 0
79680, remainder 6
87960, remainder 5
89760, remainder 6
97680, remainder 2
98760, remainder 4

That didn't work, so try numbers starting with 12346; this is impossible
because the tens digit must be 8, and the hundreds digit cannot be even.
Now try 12347, and 1234700000 has remainder 2.  The last five digits can
be

58960, remainder 6
59680, remainder 5, so this works, and the number is

1234759680.

==> arithmetic/digits/equations/123456789.p <==
In how many ways can "." be replaced with "+", "-", or "" (concatenate) in
.1.2.3.4.5.6.7.8.9=1 to form a correct equation?

==> arithmetic/digits/equations/123456789.s <==
1-2 3+4 5+6 7-8 9 = 1
+1-2 3+4 5+6 7-8 9 = 1
1+2 3+4-5+6 7-8 9 = 1
+1+2 3+4-5+6 7-8 9 = 1
-1+2 3-4+5+6 7-8 9 = 1
1+2 3-4 5-6 7+8 9 = 1
+1+2 3-4 5-6 7+8 9 = 1
1-2 3-4+5-6 7+8 9 = 1
+1-2 3-4+5-6 7+8 9 = 1
1-2-3-4 5+6 7-8-9 = 1
+1-2-3-4 5+6 7-8-9 = 1
1+2-3 4+5 6-7-8-9 = 1
+1+2-3 4+5 6-7-8-9 = 1
-1+2 3+4+5-6-7-8-9 = 1
-1 2+3 4-5-6+7-8-9 = 1
1+2+3+4-5+6+7-8-9 = 1
+1+2+3+4-5+6+7-8-9 = 1
-1+2+3-4+5+6+7-8-9 = 1
1-2-3+4+5+6+7-8-9 = 1
+1-2-3+4+5+6+7-8-9 = 1
1+2 3+4 5-6 7+8-9 = 1
+1+2 3+4 5-6 7+8-9 = 1
1+2 3-4-5-6-7+8-9 = 1
+1+2 3-4-5-6-7+8-9 = 1
1+2+3+4+5-6-7+8-9 = 1
+1+2+3+4+5-6-7+8-9 = 1
-1+2+3+4-5+6-7+8-9 = 1
1-2+3-4+5+6-7+8-9 = 1
+1-2+3-4+5+6-7+8-9 = 1
-1-2-3+4+5+6-7+8-9 = 1
1-2+3+4-5-6+7+8-9 = 1
+1-2+3+4-5-6+7+8-9 = 1
1+2-3-4+5-6+7+8-9 = 1
+1+2-3-4+5-6+7+8-9 = 1
-1-2+3-4+5-6+7+8-9 = 1
-1+2-3-4-5+6+7+8-9 = 1
-1+2 3+4 5-6 7-8+9 = 1
1-2 3-4 5+6 7-8+9 = 1
+1-2 3-4 5+6 7-8+9 = 1
-1+2 3-4-5-6-7-8+9 = 1
-1+2+3+4+5-6-7-8+9 = 1
1-2+3+4-5+6-7-8+9 = 1
+1-2+3+4-5+6-7-8+9 = 1
1+2-3-4+5+6-7-8+9 = 1
+1+2-3-4+5+6-7-8+9 = 1
-1-2+3-4+5+6-7-8+9 = 1
1+2-3+4-5-6+7-8+9 = 1
+1+2-3+4-5-6+7-8+9 = 1
-1-2+3+4-5-6+7-8+9 = 1
-1+2-3-4+5-6+7-8+9 = 1
1-2-3-4-5+6+7-8+9 = 1
+1-2-3-4-5+6+7-8+9 = 1
1-2 3+4+5+6+7-8+9 = 1
+1-2 3+4+5+6+7-8+9 = 1
1+2+3+4 5-6 7+8+9 = 1
+1+2+3+4 5-6 7+8+9 = 1
1 2+3 4+5-6 7+8+9 = 1
+1 2+3 4+5-6 7+8+9 = 1
1+2+3-4-5-6-7+8+9 = 1
+1+2+3-4-5-6-7+8+9 = 1
-1+2-3+4-5-6-7+8+9 = 1
1-2-3-4+5-6-7+8+9 = 1
+1-2-3-4+5-6-7+8+9 = 1
-1-2-3-4-5+6-7+8+9 = 1
-1-2 3+4+5+6-7+8+9 = 1
1-2+3 4-5 6+7+8+9 = 1
+1-2+3 4-5 6+7+8+9 = 1
1 2-3 4+5-6+7+8+9 = 1
+1 2-3 4+5-6+7+8+9 = 1
Total solutions  = 69 (26 of which have a leading "+", which is redundant)

69/19683 = 0.35 %

for those who care (it's not very elegant but it did the trick):

#include <stdio.h>
#include <math.h>

main (argc,argv)
int argc;
char *argv[];
{
int sresult, result, operator,thisop;
char buf,ops;
int i,j,tot=0,temp;

ops = ' ';
ops = '-';
ops = '+';

for (i=1; i<10; i++) operator[i] = 0;

for (j=0; j < 19683; j++) {
result = 0;
sresult = 0;
thisop = 1;
for (i=1; i<10; i++) {
switch (operator[i]) {
case 0:
sresult = sresult * 10 + i;
break;
case 1:
result = result + sresult * thisop;
sresult = i;
thisop = -1;
break;
case 2:
result = result + sresult * thisop;
sresult = i;
thisop = 1;
break;
}
}

result  = result + sresult * thisop;
if (result == 1) {
tot++;
for  (i=1;i<10;i++)
printf("%c%d",ops[operator[i]],i);
printf(" = %d\n",result);
}
temp = 0;
operator += 1;
for (i=1;i<10;i++) {
operator[i] += temp;
if (operator[i] > 2) { operator[i] = 0; temp = 1;}
else temp = 0;
}

}

printf("Total solutions  = %d\n" , tot);
}

cwren@media.mit.edu (Christopher Wren)

==> arithmetic/digits/equations/1992.p <==
1 = -1+9-9+2.  Extend this list to 2 through 100 on the left side of
the equals sign.

==> arithmetic/digits/equations/1992.s <==
1 = -1+9-9+2
2 = 1*9-9+2
3 = 1+9-9+2
4 = 1+9/9+2
5 = 1+9-sqrt(9)-2
6 = 1^9+sqrt(9)+2
7 = -1+sqrt(9)+sqrt(9)+2
8 = 19-9-2
9 = (1/9)*9^2
10= 1+(9+9)/2
11= 1+9+sqrt(9)-2
12= 19-9+2
13= (1+sqrt(9))!-9-2
14= 1+9+sqrt(9)!-2
15= -1+9+9-2
16= -1+9+sqrt(9)!+2
17= 1+9+9-2
18= 1+9+sqrt(9)!+2
19= -1+9+9+2
20= (19-9)*2
21= 1+9+9+2
22= (-1+sqrt(9))*(9-2)
23= (1+sqrt(9))!-sqrt(9)+2
24= -1+9*sqrt(9)-2
25= 1*9*sqrt(9)-2
26= 19+9-2
27= 1*9+9*2
28= 1+9+9*2
29= 1*9*sqrt(9)+2
30= 19+9+2
31= (1+sqrt(9))!+9-2
32= -1+sqrt(9)*(9+2)
33= 1*sqrt(9)*(9+2)
34= (-1+9+9)*2
35= -1+(9+9)*2
36= 1^9*sqrt(9)!^2
37= 19+9*2
38= 1*sqrt(9)!*sqrt(9)!+2
39= 1+sqrt(9)!*sqrt(9)!+2
40= (1+sqrt(9)!)*sqrt(9)!-2
41= -1+sqrt(9)!*(9-2)
42= (1+sqrt(9))!+9*2
43= 1+sqrt(9)!*(9-2)
44= -1+9*(sqrt(9)+2)
45= 1*9*(sqrt(9)+2)
46= 1+9*(sqrt(9)+2)
47= (-1+sqrt(9)!)*9+2
48= 1*sqrt(9)!*(sqrt(9)!+2)
49= (1+sqrt(9)!)*(9-2)
50= (-1+9)*sqrt(9)!+2
51= -1+9*sqrt(9)!-2
52= 1*9*sqrt(9)!-2
53= -1+9*sqrt(9)*2
54= 1*9*sqrt(9)*2
55= 1+9*sqrt(9)*2
56= 1*9*sqrt(9)!+2
57= 1+9*sqrt(9)!+2
58= (1+9)*sqrt(9)!-2
59= 19*sqrt(9)+2
60= (1+9)*sqrt(9)*2
61= (1+sqrt(9)!)*9-2
62= -1+9*(9-2)
63= 1*9*(9-2)
64= 1+9*(9-2)
65= (1+sqrt(9)!)*9+2
66= 1*sqrt(9)!*(9+2)
67= 1+sqrt(9)!*(9+2)
68= -(1+sqrt(9))!+92
69= (1+sqrt(9))!+(9/.2)
70= (1+9)*(9-2)
71= -1-9+9^2
72= (1+sqrt(9))*9*2
73= -19+92
74= (-1+9)*9+2
75= -1*sqrt(9)!+9^2
76= 1-sqrt(9)!+9^2
77= (1+sqrt(9)!)*(9+2)
78= -1+9*9-2
79= 1*9*9-2
80= 1+9*9-2
81= 1*9*sqrt(9)^2
82= -1+9*9+2
83= 1*9*9+2
84= 1+9*9+2
85= -1-sqrt(9)!+92
86= -1*sqrt(9)!+92
87= 1-sqrt(9)!+92
88= (1+9)*9-2
89= -1*sqrt(9)+92
90= 1-sqrt(9)+92
91= -1^9+92
92= (1+9)*9+2
93= 1^9+92
94= -1+sqrt(9)+92
95= 19*(sqrt(9)+2)
96= -1+99-2
97= 1*99-2
98= 1+99-2
99= 1*9*(9+2)
100= -1+99+2

==> arithmetic/digits/equations/24.p <==
Form an expression that evaluates to 24 that contains two 3's, two 7's,
and zero or more of the operators +, -, *, and /, and parentheses.  What
about two 4's and two 7's, or three 5's and one 1, or two 3's and two 8's?

==> arithmetic/digits/equations/24.s <==
7*(3+3/7)
7*(4-4/7)
5*(5-1/5)
8/(3-8/3)

==> arithmetic/digits/equations/383.p <==
Make 383 out of 1,2,25,50,75,100 using +,-,*,/.

==> arithmetic/digits/equations/383.s <==
You can get 383 with ((2+50)/25+1)*100+75.

Of course, if you expect / as in C, the above expression is just 375.
But then you can get 383 with (25*50-100)/(1+2).  Pity there's no way
to use the 75.

If we had a language that rounded instead of truncating, we could use
((1+75+100)*50)/(25-2) or (2*75*(25+100))/(50-1).

I imagine your problem lies in not dividing things that aren't
divisible.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil

==> arithmetic/digits/equations/find.p <==
Write a program for finding expressions built out of given numbers and using
given operators that evaluate to a given value, or listing all possible values.

==> arithmetic/digits/equations/find.s <==
As set up, it requires recompilation for different sets of numbers;
it's currently set up for 8,8,3,3; to try in other numbers, stick 'em
in the 'val' array.  To find all target numbers for which the equation
is valid, uncomment the 't' loop and 'target = t', and extend the range
to be checked... you might want to turn off VERBOSE.  I don't bother
with eliminating symmetries if equal vals are given (like 8 8 3 3), so
I normally use it like

numop 24 | sort | uniq

As it stands, this gives the output:

8 / (3 - (8 / 3)) = 24.0
8 / (3 - (8 / 3)) = 24.0
8 / (3 - (8 / 3)) = 24.0
8 / (3 - (8 / 3)) = 24.0

As you can see, there are five different kinds of binary trees with
exactly four leaf nodes.  The program tries all four operators in each
place, and all four values in each of the leaves, guaranteeing that each
is used only once... a fairly quick operation.  A small extract from
'numop 1' shows the five different shapes of trees:

((3 * 8) / 3) / 8 = 1.0
(3 * (8 / 3)) / 8 = 1.0
(3 - 3) + (8 / 8) = 1.0
3 * ((8 / 3) / 8) = 1.0
3 * (8 / (3 * 8)) = 1.0

Probably FUDGE ought to be set a little lower, for more confidence that
the equality isn't fortuitous.  Extensions to other binary operators are
obvious; unary operators and more values are not.  For a more general
problem I'd go recursive, use exact rational arithmetic, and have a fine
old time.

Enjoy...

Jim Gillogly <uunet!rand.org!James_Gillogly>
21 Wedmath S.R. 1993, 10:58
----------------------------------------------------------------

/* numop: using elementary operations on 4 numbers, find a
* desired result; e.g. 24.
*
* Don't worry about symmetries resulting in multiple correct answers.
*
* 11 Aug 93, SCRYER
*/

#include <stdio.h>

#define VERBOSE

#define MUL 0
#define DIV 1
#define ADD 2
#define SUB 3

#define FUDGE 0.01

float val = {8, 8, 3, 3};
float eval(), atof(), fabs();
char nameop();

int divzero;

main(argc, argv)
int argc;
char *argv[];
{
int op1, op2, op3;
int iv1, iv2, iv3, iv4;
int used;
int i;
float target;
float e1, e2, e3;
int t, winner;

if (argc != 2)
{
fprintf(stderr, "Usage: numop <target>\n");
exit(1);
}
target = atof(argv);

/* for (t = -1000; t < 1000; t++) */
{
/*    target = t;*/
winner = 0;

for (i = 0; i < 4; i++) used[i] = 0;

for (op1 = 0; op1 < 4; op1++)
for (op2 = 0; op2 < 4; op2++)
for (op3 = 0; op3 < 4; op3++)
for (iv1 = 0; iv1 < 4; iv1++)
{
used[iv1] = 1;
for (iv2 = 0; iv2 < 4; iv2++)
{
if (used[iv2]) continue;
used[iv2] = 1;
for (iv3 = 0; iv3 < 4; iv3++)
{
if (used[iv3]) continue;
used[iv3] = 1;
for (iv4 = 0; iv4 < 4; iv4++)
{
if (used[iv4]) continue;

/* Case 1 */
divzero = 0;
e3 = eval(op3, val[iv3], val[iv4]);
e2 = eval(op2, val[iv1], val[iv2]);
e1 = eval(op1, e2, e3);                         /* (u + v) * (w - x) */
if (fabs(e1 - target) < FUDGE && ! divzero)
#ifdef VERBOSE
printf("(%.0f %c %.0f) %c (%.0f %c %.0f) = %.1f\n",
val[iv1], nameop(op2), val[iv2], nameop(op1),
val[iv3], nameop(op3), val[iv4], e1);
#else
winner = 1;
#endif
/* Case 2 */
divzero = 0;
e3 = eval(op3, val[iv1], val[iv2]);
e2 = eval(op2, e3, val[iv3]);
e1 = eval(op1, e2, val[iv4]);                   /* ((u + v) * w) - x */
if (fabs(e1 - target) < FUDGE && ! divzero)
#ifdef VERBOSE
printf("((%.0f %c %.0f) %c %.0f) %c %.0f = %.1f\n",
val[iv1], nameop(op3), val[iv2], nameop(op2), val[iv3], nameop(op1), val[iv4], e1);
#else
winner = 1;
#endif

/* Case 3 */
divzero = 0;
e3 = eval(op3, val[iv2], val[iv3]);
e2 = eval(op2, val[iv1], e3);
e1 = eval(op1, e2, val[iv4]);                   /* (u + (v * w)) - x */
if (fabs(e1 - target) < FUDGE && ! divzero)
#ifdef VERBOSE
printf("(%.0f %c (%.0f %c %.0f)) %c %.0f = %.1f\n",
val[iv1], nameop(op2), val[iv2], nameop(op3), val[iv3],
nameop(op1), val[iv4], e1);
#else
winner = 1;
#endif

/* Case 4 */
divzero = 0;
e3 = eval(op3, val[iv2], val[iv3]);
e2 = eval(op2, e3, val[iv4]);
e1 = eval(op1, val[iv1], e2);                   /* u + ((v * w) - x) */
if (fabs(e1 - target) < FUDGE && ! divzero)
#ifdef VERBOSE
printf("%.0f %c ((%.0f %c %.0f) %c %.0f) = %.1f\n",
val[iv1], nameop(op1), val[iv2], nameop(op3), val[iv3],
nameop(op2), val[iv4], e1);
#else
winner = 1;
#endif

/* Case 5 */                                    /* u + (v * (w - x)) */
divzero = 0;
e3 = eval(op3, val[iv3], val[iv4]);
e2 = eval(op2, val[iv2], e3);
e1 = eval(op1, val[iv1], e2);
if (fabs(e1 - target) < FUDGE && ! divzero)
#ifdef VERBOSE
printf("%.0f %c (%.0f %c (%.0f %c %.0f)) = %.1f\n",
val[iv1], nameop(op1), val[iv2], nameop(op2), val[iv3],
nameop(op3), val[iv4], e1);
#else
winner = 1;
#endif

}
used[iv3] = 0;
}
used[iv2] = 0;
}
used[iv1] = 0;
}
#ifndef VERBOSE
if (winner) printf("%d\n", t), fflush(stdout);
#endif
}
}

char nameop(op)
int op;
{
switch(op)
{
case MUL: return '*';
case DIV: return '/';
case ADD: return '+';
case SUB: return '-';
}
return '?';
}

float eval(op, val1, val2)
int op;
float val1, val2;
{
switch(op)
{
case MUL: return val1 * val2;
case DIV:
if (val2 == 0.)
{
divzero = 1;
#ifdef EXTREMELYVERBOSE
fprintf(stderr, "Division by zero.\n");
#endif
}
return val2 == 0.? 0. : val1 / val2;
case ADD: return val1 + val2;
case SUB: return val1 - val2;
}
return 0.;
}

==> arithmetic/digits/extreme.products.p <==
What are the extremal products of three three-digit numbers using digits 1-9?

==> arithmetic/digits/extreme.products.s <==
There is a simple procedure which applies to these types of problems (and
which can be proven with help from the arithmetic-geometric inequality).

For the first part we use the "first large then equal" procedure.
This means that are three numbers will be 7**, 8**, and 9**.  Now
the digits 4,5,6 get distributed so as to make our three number as
close to each other as possible, i.e. 76*, 85*, 94*.  The same goes
for the remaining three digits, and we get 763, 852, 941.

For the second part we use the "first small then different" procedure.
Our three numbers will be of the form 1**, 2**, 3**.  We now place
the three digits so as to make our three numbers as unequal as possible;
this gives 14*, 25*, 36*.  Finishing, we get 147, 258, 369.

Now, *prove* that these procedures work for generalizations of this
problem.

==> arithmetic/digits/labels.p <==
You have an arbitrary number of model kits (which you assemble for
fun and profit).  Each kit comes with twenty (20) stickers, two of which
are labeled "0", two are labeled "1", ..., two are labeled "9".
You decide to stick a serial number on each model you assemble starting
with one.  What is the first number you cannot stick.  You may stockpile
unused numbers on already assembled models, but you may not crack open
a new model to get at its stickers.  You complete assembling the current
model before starting the next.

==> arithmetic/digits/labels.s <==
The method I used for this problem involved first coming up with a
formula that says how many times a digit has been used in all n models.

n = k*10^i + m for some k,m with 0 <k <10, m < 10^i
f(d,n) = (number of d's used getting to k*10^i from digits 0 to (i-1)) +
(number of d's used by #'s 10^i to n from digit i) + f(d,m)
f(d,n) = i*k*10^(i-1) + (if (d < k) 10^i else if (d == k) m+1 else 0) + f(d,m)

This doesn't count 0's, which should be ok as they are not used as often
as other digits.  From the formula, it is clear that f(1,n) is never
less than f(d,n) for 1<d<10.
So I just calculated f(1,n) for various n (with some help from bc).

I quickly discovered that for n = 2*10^15, f(1,n) = 2*n.  After further
trials I determined that for n = 1999919999999981, f(1,n) = 2*n + 1.
This appears to be the smallest n with f(1,n) > 2*n.

==> arithmetic/digits/least.significant/factorial.p <==
What is the least significant non-zero digit in the decimal expansion of n!?

==> arithmetic/digits/least.significant/factorial.s <==
Reduce mod 10 the numbers 2..n and then cancel out the
required factors of 10. The final step then involves
computing 2^i*3^j*7^k mod 10 for suitable i,j and k.

A small program that performs this calculation is appended. Like the
other solutions, it takes O(log n) arithmetic operations.

-kym
===

#include<stdio.h>
#include<assert.h>

int	p={
/*2*/	2,	4,	8,	6,
/*3*/	3,	9,	7,	1,
/*4*/	4,	6,	4,	6,
/*5*/	5,	5,	5,	5,
/*6*/	6,	6,	6,	6,
/*7*/	7,	9,	3,	1,
};

main(){
int	i;
int n;

for(n=2;n<1000;n++){
i=lsdfact(n);
printf("%d\n",i);
}

exit(0);
}

lsdfact(n){
int	a;
int	i;
int	n5;
int	tmp;

for(i=0;i<=9;i++)a[i]=alpha(i,n);

n5=0;
/* NOTE: order is important in following */
l5:;
while(tmp=a){	/* cancel factors of 5 */
n5+=tmp;
a+=(tmp+4)/5;
a+=(tmp+3)/5;
a=(tmp+2)/5;
a+=(tmp+1)/5;
a+=(tmp+0)/5;
}
l10:;
if(tmp=a){
a=0;	/* cancel all factors of 10 */
for(i=0;i<=9;i++)a[i]+=alpha(i,tmp);
}
if(a) goto l5;
if(a) goto l10;

/* n5 == number of 5's cancelled;
must now cancel same number of factors of 2 */
i=ipow(2,a+2*a+a+3*a-n5)*
ipow(3,a+a+2*a)*
ipow(7,a);
assert(i%10);	/* must not be zero */
return	i%10;
}

alpha(d,n){
/* number of decimal numbers in [1,n] ending in digit d */
int tmp;
tmp=(n+10-d)/10;
if(d==0)tmp--;	/* forget 0 */
return tmp;
}

ipow(x,y){
/* x^y mod 10 */
if(y==0) return 1;
if(y==1) return x;
return p[x-2][(y-1)%4];
}

==> arithmetic/digits/least.significant/tower.of.power.p <==
What are the least significant digits of 9^(8^(7^(6^(5^(4^(3^(2^1))))))) ?

==> arithmetic/digits/least.significant/tower.of.power.s <==
9^11 = 9 (mod 100), so we need to find 8^...^1 (mod 10).
8^5 = 8 (mod 10), so we need to find 7^...^1 (mod 4).
7^3 = 7 (mod 4), so we need to find 6^...^1 (mod 2), but
this is clearly 0, so 7^...^1 = 1 (mod 4) ==>
8^...^1 = 8 (mod 10) ==> 9^...^1 = 9^8 (mod 100) = 21 (mod 100).

==> arithmetic/digits/most.significant/googol.p <==
What digits does googol! start with?

==> arithmetic/digits/most.significant/googol.s <==
I'm not sure how to calculate the first googol of digits of log10(e), but
here's the first 150(approximately) of them...

0.43429448190325182765112891891660508229439700580366656611445378316586464920
8870774729224949338431748318706106744766303733641679287158963906569221064663

We need to deal with the digits immediately after the decimal point in
googol*log10(e), which are  .187061

frac[log(googol!)] = frac[halflog2pi + 50 + googol(100-log10(e))]
= frac{halflog2pi + frac[googol(100-log10(e))]}
= frac[.399090 + (1- .187061)]
= .212029

10 ** .212029 = 1.629405

Which means that googol! starts with 1629

==> arithmetic/digits/most.significant/powers.p <==
What is the probability that 2^N begins with the digits 603245?

==> arithmetic/digits/most.significant/powers.s <==
2^N begins with 603245 iff 603246*10^m > 2^N >= 603245*10^m for some
positive integer m ==> m+log(603246) > N*log(2) >= m+log(603245);
so 2^N begins with 603245 iff frac(log(603246)) > frac(N*log(2))
>= frac(log(603245)).  If we are using natural density then N*log(2)
is uniformly distributed mod 1 since log(2) is irrational, hence the
probability is frac(log(603246)) - frac(log(603245)) =
frac(log(603246)-log(603245)) = frac(log(603246/603245)).

A neat observation is that since it is known p_n*c, where p_n is the
nth prime and c is irrational, is uniformly distributed mod 1, we get
the same answer if we replace 2^N with 2^{p_n}.
--
Chris Long, 265 Old York Rd., Bridgewater, NJ  08807-2618

==> arithmetic/digits/nine.digits.p <==
Form a number using 0-9 once with its first n digits divisible by n.

==> arithmetic/digits/nine.digits.s <==
First, reduce the sample set. For each digit of ABCDEFGHI, such that the last
digit, (current digit), is the same as a multiple of N :

A: Any number 1-9
B: Even numbers 2,4,6,8 (divisible by 2).
C: Any number 1-9 (21,12,3,24,15,6,27,18,9).
D: Even numbers 2,4,6,8 (divisible by 4, every other even).
E: 5 (divisible by 5 and 0 not allowed).
F: Even numbers (12,24,6,18)
G: Any number 1-9 (21,42,63,14,35,56,7,28,49).
H: Even numbers (32,24,16,8)
I: Any number 1-9 (81,72,63,54,45,36,27,18,9)

Since E must be 5, I can eliminate it everywhere else.
Since I will use up all the even digits, (2,4,6,8) filling in those spots
that must be even. Any number becomes all odds, except 5.

A: 1,3,7,9
B: 2,4,6,8
C: 1,3,7,9
D: 2,4,6,8
E: 5
F: 2,4,6,8
G: 1,3,7,9
H: 2,4,6,8
I: 1,3,7,9

We have that 2C+D=0 (mod 4), and since C is odd,
this implies that D is 2 or 6; similarly we find that H is 2 or 6 ==>
{B,F} = {4,8}.  D+5+F=0 (mod 3) ==> if D=2 then F=8, if D=6 then F=4.

We have two cases.

Assume our number is of the form A4C258G6I0.  Now the case n=8 ==>
G=1,9; case n=3 ==> A+1+C=0 (mod 3) ==> {A,C}={1,7} ==> G=9, I=3.
The two numbers remaining fail for n=7.

Assume our number is of the form A8C654G2I0.  The case n=8 ==> G=3,7.
If G=3, we need to check to see which of 1896543, 9816543, 7896543,
and 9876543 are divisible by 7; none are.

If G=7, we need to check to see which of 1896547, 9816547, 1836547,
and 3816547 are divisible by 7; only the last one is, which yields
the solution 3816547290.

==> arithmetic/digits/palindrome.p <==
Does the series formed by adding a number to its reversal always end in
a palindrome?

==> arithmetic/digits/palindrome.s <==
This is not known.

If you start with 196, after 9480000 iterations you get a 3924257-digit
non-palindromic number.  However, there is no known proof that you will
never get a palindrome.

The statement is provably false for binary numbers. Roland Sprague has
shown that 10110 starts a series that never goes palindromic.

==> arithmetic/digits/palintiples.p <==
Find all numbers that are multiples of their reversals.

==> arithmetic/digits/palintiples.s <==
We are asked to find numbers that are integer multiples of their
reversals, which I call palintiples.  Of course, all the palindromic
numbers are a trivial example, but if we disregard the unit multiples,
the field is narrowed considerably.

Rouse Ball (_Mathematical_recreations_and_essays_) originated the
problem, and G. H. Hardy (_A_mathematician's_apology_) used the result
that 9801 and 8712 are the only four-digit palintiples as an example
of a theorem that is not ``serious''.  Martin Beech (_The_mathema-
tical_gazette_, Vol 74, #467, pp 50-51, March '90) observed that
989*01 and 879*12 are palintiples, an observation he ``confirmed'' on
a hand calculator, and conjectured that these are all that exist.

I confirm that Beech's numbers are palintiples, I will show that they
are not all of the palintiples.  I will show that the palintiples do
not form a regular language.  And then I will prove that I have found
all the palintiples, by describing the them with a generalized form
of regular expression.  The results become more interesting in other
bases.

First, I have a more reasonable method of confirming that these
numbers are palintiples:

Proof:  First, letting "9*" and "0*" refer an arbitrary string of
nines and a string of zeroes of the same length, I note that

879*12 = 879*00 + 12 = (880*00 - 100) + 12 = 880*00 - 88
219*78 = 219*00 + 78 = (220*00 - 100) + 78 = 220*00 - 22

989*01 = 989*00 +  1 = (990*00 - 100) +  1 = 990*00 - 99
109*89 = 109*00 + 89 = (110*00 - 100) + 89 = 110*00 - 11

It is obvious that 4x(220*00 - 22) = 880*00 - 88 and that
9x(110*00 - 11) = 990*00 - 99.  QED.

Now, to show that these palintiples are not all that exist, let us
take the (infinite) language L = (879*12 + 0*), and let Pal(L)
refer to the set of palindromes over the alphabet L.  It is
immediate that the numbers in Pal(L) are palintiples.  For
instance,

8712 000 87912 879999912 879999912 87912 000 8712
= 4 x 2178 000 21978 219999978 219999978 21978 000 2178

(where I have inserted spaces to enhance readability) is a palintiple.
Similarly, taking L = (989*01 + 0*), the numbers in Pal(L) are
palintiples.  We exclude numbers starting with zeroes.

The reason these do not form a regular language is that the
sub-palintiples on the left end of the number must be the same (in
reverse order) as the sub-palintiples on the right end of the number:

8712 8712 87999912 = 4 x 2178 2178 21999978

is not a palintiple, because 8712 8712 87999912 is not the reverse of
2178 2178 21999978.  The pumping lemma can be used to prove that
Pal(L)+Pal(L) is not a regular language, just as in the familiar
proof that the palindromes over a non-singleton alphabet do not form a
regular language.

Now to characterize all the palintiples, let N be a palintiple,
N=CxR(N), where R(.) signifies reversal, and C>1 is an integer.  (I
use "x" for multiplication, to avoid confusion with the Kleene star
"*", which signifies the concatenated closure.)  If D is a digit of N,
let D' refer to the corresponding digit of R(N).  Since N=CxR(N),
D+10T = CxD'+S, where S is the carry in to the position occupied by D'
when R(N) is multiplied by C, and T is the carry out of that position.
Similarly, D'+10T'=CxD+S', where S', T' are carries in and out of the
position occupied by D when R(N) is multiplied by C.

Since D and D' are so closely related, I will use the symbol D:D' to
refer to a digit D on the left side of a string with a corresponding
digit D' on the right side of the string.  More formally, an
expression "x:y x:y ... x[n]:y[n] w" will refer to a
string "x x ... x[n] w y[n] ... y y", where the x[i] and
y[i] are digits and w is a string of zero or one digits.  So 989901
may be written as 9:1 8:0 9:9 and 87912 may be written as 8:2 7:1 9.
Thus Pal(L)+Pal(L) (omitting numbers with leading zeroes) can be
represented as

(8:2 7:1 9:9* 1:7 2:8 0:0*)*
(0:0* + 0 + 8:2 7:1 ( 9:9* + 9:9* 9))
+ (9:1 8:0 9:9* 0:8 1:9 0:0*)*
(0:0* + 0 + 9:1 8:0 ( 9:9* + 9:9* 9)).      (1)

For each pair of digits D:D', there are a very limited--and often
empty--set of quadruples S,T,S',T' of digits that satisfy the
equations

D +10T =CxD'+S
D'+10T'=CxD +S',                      (2)

yet such a quadruple must exist for "D:D'" to appear in a palintiple
with multiplier C.  Furthermore, the S and T' of one D:D' must be T
and S', respectively, of the next pair of digits that appear.  This
enables us to construct a finite state machine to recognize those
palintiples.  The states [X#Y] refer to a pair of carries in D and D',
and we allow a transition from state [T#S'] to state [S#T'] on input
symbol D:D' exactly when equations (2) are satisfied.  Special
transitions for a single-digit input symbol (the central digit of
odd-length palintiples) and the criteria for the initial and the
accepting states are left as exercises.  The finite state machines
thus formed are

State         Symbol  New     Symbol  New      Symbol   New
Accept?         State           State             State

--> [0#0]  Y       8:2  [0#3]      0:0  [0#0]         0   [A]
[0#3]  N       7:1  [3#3]
[3#3]  Y       1:7  [3#0]      9:9  [3#3]         9   [A]
[3#0]  N       2:8  [0#0]
[A]   Y

for constant C=4, and

State         Symbol  New     Symbol  New      Symbol   New
Accept?         State           State             State

--> [0#0]  Y       1:9  [0#8]      0:0  [0#0]         0   [A]
[0#8]  N       8:0  [8#8]
[8#8]  Y       0:8  [8#0]      9:9  [8#8]         9   [A]
[8#0]  N       9:1  [0#0]
[A]   Y

for constant C=9, and the finite state machines for other constants
accept only strings of zeroes.  It is not hard to verify that the
proposed regular expression (1) represents the union of the languages
accepted by these machines, omitting the empty string and strings
beginning with zero.

I have written a computer program that constructs finite state
machines for recognizing palintiples for various bases and constants.
I found that base 10 is actually an unusually boring base for this
problem.  For instance, the machine for base 8, constant C=5 is

State         Symbol  New     Symbol  New      Symbol  New
Accept?         State           State            State

--> [0#0]  Y       0:0  [0#0]      5:1  [0#3]         0  [A]
[0#3]  N       1:0  [1#1]      6:1  [1#4]
[1#1]  Y       0:1  [3#0]      5:2  [3#3]
[3#0]  N       1:5  [0#0]      6:6  [0#3]         6  [A]
[3#3]  Y       2:5  [1#1]      7:6  [1#4]
[1#4]  N       1:1  [4#1]      6:2  [4#4]         1  [A]
[4#4]  Y       2:6  [4#1]      7:7  [4#4]         7  [A]
[4#1]  N       1:6  [3#0]      6:7  [3#3]
[A]   Y

for which I invite masochists to write the regular expression.  If
anyone wants more, I should remark that the base 29 machine for
constant C=18 has 71 states!

By the way, I did not find any way of predicting the size or form of
the machines for the various bases, except that the machines for C=B-1
all seem to be isomorphic to each other.  If anyone investigates the
general behavior, I would be most happy to hear about it.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil
May, 1992
[ A preliminary version of this message appeared in April, 1991. ]
================================================================
Dan

==> arithmetic/digits/power.two.p <==
Prove that for any 9-digit number (base 10) there is an integral power
of 2 whose first 9 digits are that number.

==> arithmetic/digits/power.two.s <==
Let v = log to base 10 of 2.
Then v is irrational.

Let w = log to base 10 of these 9 digits.

Since v is irrational, given epsilon > 0, there exists some natural number
n such that

{w} < {nv} < {w} + epsilon

({x} is the fractional part of x.)  Let us pick n for when

epsilon = log 1.00000000000000000000001.

Then 2^n does the job.

==> arithmetic/digits/prime/101.p <==
How many primes are in the sequence 101, 10101, 1010101, ...?

==> arithmetic/digits/prime/101.s <==
Note that the sequence
101 , 10101, 1010101, ....
can be viewed as
100**1 +1, 100**2 + 100**1 + 1, 100**3 + 100**2 + 100**1 +1 ....
that is,
the k-th term in the sequence is
100**k + 100**(k-1) + 100**(k-2) + ...+ 100**(1) + 1
= (100)**(k+1) - 1
----------------
11 * 9
= (10)**(2k+2) - 1
----------------
11 * 9
= ((10)**(k+1) - 1)*((10)**(k+1) +1)
---------------------------------
11*9
thus either 11 and 9 divide the numerator. Either they both divide the
same factor in the numerator or different factors in the numerator. In
any case, after dividing, they leave the numerators as a product of two
integers.  Only in the case of k = 1, one of the integers is 1. Thus
there is exactly one prime in the above sequence: 101.

==> arithmetic/digits/prime/all.prefix.p <==
What is the longest prime whose every proper prefix is a prime?

==> arithmetic/digits/prime/all.prefix.s <==
23399339, 29399999, 37337999, 59393339, 73939133

==> arithmetic/digits/prime/change.one.p <==
What is the smallest number that cannot be made prime by changing a single
digit?  Are there infinitely many such numbers?

==> arithmetic/digits/prime/change.one.s <==
200.  Obviously, you would have to change the last digit, but 201, 203,
207, and 209 are all composite.  For any smaller number, you can change
the last digit, and get
2,11,23,31,41,53,61,71,83,97,101,113,127,131,149,151,163,173,181, or 191.

200+2310n gives an infinite family, because changing the last
digit to 1 or 7 gives a number divisible by 3; to 3, a number divisible
by 7; to 9, a number divisible by 11.

==> arithmetic/digits/prime/prefix.one.p <==
2 is prime, but 12, 22, ..., 92 are not.  Similarly, 5 is prime
whereas 15, 25, ..., 95 are not.  What is the next prime number
which is composite when any digit is prefixed?

==> arithmetic/digits/prime/prefix.one.s <==
149

==> arithmetic/digits/reverse.p <==
Is there an integer that has its digits reversed after dividing it by 2?

==> arithmetic/digits/reverse.s <==
Assume there's such a positive integer x such that x/2=y and y is the
reverse of x.

Then x=2y.  Let x = a...b, then y = b...a, and:

b...a   (y)
x     2
--------
a...b   (x)

From the last digit b of x, we have b = 2a (mod 10), the possible
values for b are 2, 4, 6, 8 and hence possible values for (a, b) are
(1,2), (6,2), (2,4), (7,4), (3,6), (8,6), (4,8), (9,8).

From the first digit a of x, we have a = 2b or a = 2b+1.  None of the
above pairs satisfy this condition.  A contradiction.

Hence there's no such integer.

==> arithmetic/digits/rotate.p <==
Find integers where multiplying them by single digits rotates their digits
one position, so that the last digit become the first digit.

==> arithmetic/digits/rotate.s <==
2 105263157894736842
3 1034482758620689655172413793
4 102564 153846 179487 205128 230769
5 142857 102040816326530612244897959183673469387755
6 1016949152542372881355932203389830508474576271186440677966
1186440677966101694915254237288135593220338983050847457627
1355932203389830508474576271186440677966101694915254237288
1525423728813559322033898305084745762711864406779661016949
7 1014492753623188405797 1159420289855072463768 1304347826086956521739
8 1012658227848 1139240506329
9 10112359550561797752808988764044943820224719

In base B, suppose you have an N-digit answer A whose digits are
rotated when multiplied by K.  If D is the low-order digit of A, we
have

(A-D)/B + D B^(N-1) = K A .

Solving this for A we have

D (B^N - 1)
A = ----------- .
B K - 1

In order for A >= B^(N-1) we must have D >= K.  Now we have to find N
such that B^N-1 is divisible by R=(BK-1)/gcd(BK-1,D).  This always has
a minimal solution N0(R,B)<R, and the set of all solutions is the set
of multiples of N0(R,B).  N0(R,B) is the length of the repeating part
of the fraction 1/R in base B.

N0(ST,B)=N0(S,B)N0(T,B) when (S,T)=1, and for prime powers, N0(P^X,B)
divides (P-1)P^(X-1). Determining which divisor is a little more
complicated but well-known (cf. Hardy & Wright).

So given B and K, there is one minimal solution for each
D=K,K+1,...,B-1, and you get all the solutions by taking repetitions
of the minimal solutions.

==> arithmetic/digits/sesqui.p <==
Find the least number where moving the first digit to the end multiplies by 1.5.

==> arithmetic/digits/sesqui.s <==
Let's represent this number as  a*10^n+b,  where 1<=a<=9 and
b < 10^n.  Then the condition to be satisfied is:

3/2(a*10^n+b) = 10b+a

3(a*10^n+b) = 20b+2a

3a*10^n+3b = 20b+2a

(3*10^n-2)a = 17b

b = a*(3*10^n-2)/17

So we must have 3*10^n-2 = 0 (mod 17) (since a is less than 10, it
cannot contribute the needed prime 17 to the factorization of 17b).
(Also, assuming large n, we must have a at most 5 so that b < 10^n will
be satisfied, but note that we can choose a=1).  Now,

3*10^n-2 = 0 (mod 17)

3*10^n = 2 (mod 17)

10^n = 12 (mod 17)

A quick check shows that the smallest n which satisfies this is 15
(the fact that one exists was assured to us because 17 is prime).  So,
setting n=15 and a=1 (obviously) gives us b=176470588235294, so the
number we are looking for is

1176470588235294

and, by the way, we can set a=2 to give us the second smallest such
number,
2352941176470588

Other things we can infer about these numbers is that there are 5 of
them less than 10^16, 5 more less than 10^33, etc.

==> arithmetic/digits/squares/change.leading.p <==
What squares remain squares when their leading digits are incremented?

==> arithmetic/digits/squares/change.leading.s <==
Omitting solutions that are obtained from smaller solutions by
multiplying by powers of 10, the squares of these numbers satisfy the
condition:

1. (105,145), (3375,4625), (14025,17225), (326625,454625),
(10846875,14753125), (43708125,53948125), ...

2. (45,55), (144375,175625), (463171875,560828125), ...

7. (2824483699753370361328125,2996282391593370361328125), ...

Here is how to find them. We have (y+x)*(y-x) = 10^n, and so we must have
{y+x, y-x} as {5^m*10^a, 2^m*10^b} in some order. It is also necessary (and
sufficient) that y/x lies in the interval [sqrt(3/2),sqrt(2)], or equivalently
that (y+x)/(y-x) lies in [3+sqrt(8),5+sqrt(24)] = [5.82842...,9.89897...].
Thus we need to make (5/2)^m*10^(a-b), or its reciprocal, in this range.
For each m there is clearly at most one power of 10 that will do. m=2,a=b
gives (105,145); m=3,b=a+2 gives (3375,4625), and so on.

There are infinitely many non-equivalent solutions, because log(5/2) / log(10)
is irrational.

One can use exactly the same argument to find squares whose initial 2 can
be replaced by a 3, of course, except that the range of (y+x)/(y-x) changes.

==> arithmetic/digits/squares/length.22.p <==
Is it possible to form two numbers A and B from 22 digits such that
A = B^2?  Of course, leading digits must be non-zero.

==> arithmetic/digits/squares/length.22.s <==
No, the number of digits of A^2 must be of the form 3n or 3n-1.

==> arithmetic/digits/squares/length.9.p <==
Is it possible to make a number and its square, using the digits from 1
through 9 exactly once?

==> arithmetic/digits/squares/length.9.s <==
Yes, there are two such pairs: (567, 567^2=321489) and (854,854^2=729316).

```

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