Archive-name: puzzles/archive/logic/part1
Last-modified: 17 Aug 1993
Version: 4
See reader questions & answers on this topic! - Help others by sharing your knowledge ==> logic/29.p <== Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each, totalling $27. The bellboy has $2, totalling $29. Where is the remaining dollar? ==> logic/29.s <== Each person paid $9, totalling $27. The manager has $25 and the bellboy $2. The bellboy's $2 should be added to the manager's $25 or subtracted from the tenants' $27, not added to the tenants' $27. ==> logic/ages.p <== 1) Ten years from now Tim will be twice as old as Jane was when Mary was nine times as old as Tim. 2) Eight years ago, Mary was half as old as Jane will be when Jane is one year older than Tim will be at the time when Mary will be five times as old as Tim will be two years from now. 3) When Tim was one year old, Mary was three years older than Tim will be when Jane is three times as old as Mary was six years before the time when Jane was half as old as Tim will be when Mary will be ten years older than Mary was when Jane was one-third as old as Tim will be when Mary will be three times as old as she was when Jane was born. HOW OLD ARE THEY NOW? ==> logic/ages.s <== The solution: Tim is 3, Jane is 8, and Mary is 15. A little grumbling is in order here, as clue number 1 leads to the situation a year and a half ago, when Tim was 1 1/2, Jane was 6 1/2, and Mary was 13 1/2. This sort of problem is easy if you write down a set of equations. Let t be the year that Tim was born, j be the year that Jane was born, m be the year that Mary was born, and y be the current year. As indefinite years come up, let y1, y2, ... be the indefinite years. Then the equations are y + 10 - t = 2 (y1 - j) y1 - m = 9 (y1 - t) y - 8 - m = 1/2 (y2 - j) y2 - j = 1 + y3 - t y3 - m = 5 (y + 2 - t) t + 1 - m = 3 + y4 - t y4 - j = 3 (y5 - 6 - m) y5 - j = 1/2 (y6 - t) y6 - m = 10 + y7 - m y7 - j = 1/3 (y8 - t) y8 - m = 3 (j - m) t = y - 3 j = y - 8 m = y - 15 ==> logic/attribute.p <== All the items in the first list share a particular attribute. The second list is of some items lacking the attribute. Set#1 with: battery, key, yeast, bookmark w/out: stapler, match, Rubik's cube, pill bottle Set#2 with: Rubik's cube, chess set, electrical wiring, compass needle w/out: clock, rope, tic-tac-toe, pencil sharpener Set#3: with: koosh, small intestine, Yorkshire Terrier, Christmas Tree w/out: toothbrush, oak chair, soccer ball, icicle Points to realize: 1. There may be exceptions to any item on the list, for instance a particular clock may share the properties of the 'with' list of problem two, BUT MOST ORDINARY clocks do not. All the properties apply the vast majority of the the items mentioned. Extraordinary exceptions should be ignored. 2. Pay the most attention to the 'with' list. The 'without' list is only present to eliminate various 'stupid' answers. ==> logic/attribute.s <== The attribute puzzle format is a traditional format in math education. It occurs in logic materials developed in the sixties by EDC in Boston, with visual objects. Example: these are gloops: A B C D E these are not gloops: F G J L N which of these are gloops? O P Q R S Set#1 with: battery, key, yeast, bookmark w/out: stapler, match, Rubik's cube, pill bottle Needs to be placed inside something else when used for its usual purpose. Set#2 with: Rubik's cube, chess set, electrical wiring, compass needle w/out: clock, rope, tic-tac-toe, pencil sharpener Uses color to distinguish between otherwise identical parts. Set#3: with: koosh, small intestine, Yorkshire Terrier, Christmas Tree w/out: toothbrush, oak chair, soccer ball, icicle Villiform. ==> logic/bookworm.p <== A bookworm eats from the first page of an encyclopedia to the last page. The bookworm eats in a straight line. The encyclopedia consists of ten 1000-page volumes and is sitting on a bookshelf in the usual order. Not counting covers, title pages, etc., how many pages does the bookworm eat through? ==> logic/bookworm.s <== On a book shelf the first page of the first volume is on the "inside" __ __ B| | | |F A|1 |...........................|10|R C| | | |O K| | | |N | | | |T ---------------------------------- so the bookworm eats only through the cover of the first volume, then 8 times 1000 pages of Volumes 2 - 9, then through the cover to the 1st page of Vol 10. He eats 8,000 pages. If the bookworm ate the first page and the last page, it ate 8,004 pages. ==> logic/boxes.p <== Which Box Contains the Gold? Two boxes are labeled "A" and "B". A sign on box A says "The sign on box B is true and the gold is in box A". A sign on box B says "The sign on box A is false and the gold is in box A". Assuming there is gold in one of the boxes, which box contains the gold? ==> logic/boxes.s <== The problem cannot be solved with the information given. The sign on box A says "The sign on box B is true and the gold is in box A". The sign on box B says "The sign on box A is false and the gold is in box A". The following argument can be made: If the statement on box A is true, then the statement on box B is true, since that is what the statement on box A says. But the statement on box B states that the statement on box A is false, which contradicts the original assumption. Therefore, the statement on box A must be false. This implies that either the statement on box B is false or that the gold is in box B. If the statement on box B is false, then either the statement on box A is true (which it cannot be) or the gold is in box B. Either way, the gold is in box B. However, there is a hidden assumption in this argument: namely, that each statement must be either true or false. This assumption leads to paradoxes, for example, consider the statement: "This statement is false." If it is true, it is false; if it is false, it is true. The only way out of the paradox is to deny that the statement is either true or false and label it meaningless instead. Both of the statements on the boxes are therefore meaningless and nothing can be concluded from them. In general, statements about the truth of other statements lead to contradictions. Tarski invented metalanguages to avoid this problem. To avoid paradox, a statement about the truth of a statement in a language must be made in the metalanguage of the language. Common sense dictates that this problem cannot be solved with the information given. After all, how can we deduce which box contains the gold simply by reading statements written on the outside of the box? Suppose we deduce that the gold is in box B by whatever line of reasoning we choose. What is to stop us from simply putting the gold in box A, regardless of what we deduced? (cf. Smullyan, "What Is the Name of This Book?", Prentice-Hall, 1978, #70) ==> logic/camel.p <== An Arab sheikh tells his two sons to race their camels to a distant city to see who will inherit his fortune. The one whose camel is slower will win. The brothers, after wandering aimlessly for days, ask a wise man for advise. After hearing the advice they jump on the camels and race as fast as they can to the city. What does the wise man say? ==> logic/camel.s <== The wiseman tells them to switch camels. ==> logic/centrifuge.p <== You are a biochemist, working with a 12-slot centrifuge. This is a gadget that has 12 equally spaced slots around a central axis, in which you can place chemical samples you want centrifuged. When the machine is turned on, the samples whirl around the central axis and do their thing. To ensure that the samples are evenly mixed, they must be distributed in the 12 slots such that the centrifuge is balanced evenly. For example, if you wanted to mix 4 samples, you could place them in slots 12, 3, 6 and 9 (assuming the slots are numbered from 1 to 12 like a clock). Problem: Can you use the centrifuge to mix 5 samples? ==> logic/centrifuge.s <== The superposition of any two solutions is yet another solution, so given that the factors > 1 of 12 (2, 3, 4, 6, 12) are all solutions, the only thing to check about, for example, the proposed solution 2+3 is that not all ways of combining 2 & 3 would have centrifuge tubes from one subsolution occupying the slot for one of the tubes in another solution. For the case 2+3, there is no problem: Place 3 tubes, one in every 4th position, then place the 4th and 5th diametrically opposed (each will end up in a slot adjacent to one of the first 3 tubes). The obvious generalization is, what are the numbers of tubes that cannot be balanced? Observing that there are solutions for 2,3,4,5,6 tubes and that if X has a solution, 12-X has also one (obtained by swapping tubes and holes), it is obvious that 1 and 11 are the only cases without solutions. Here is how this problem is often solved in practice: A dummy tube is added to produce a total number of tubes that is easy to balance. For example, if you had to centrifuge just one sample, you'd add a second tube opposite it for balance. ==> logic/chain.p <== What is the least number of links you can cut in a chain of 21 links to be able to give someone all possible number of links up to 21? ==> logic/chain.s <== Two. OOO C OOOOO C OOOOOOOOOOO (where Os are chained unbroken links, and the Cs are the unchained broken links) And equivalently: OOO C OOOOOO C OOOOOOOOOO ==> logic/children.p <== A man walks into a bar, orders a drink, and starts chatting with the bartender. After a while, he learns that the bartender has three children. "How old are your children?" he asks. "Well," replies the bartender, "the product of their ages is 72." The man thinks for a moment and then says, "that's not enough information." "All right," continues the bartender, "if you go outside and look at the building number posted over the door to the bar, you'll see the sum of the ages." The man steps outside, and after a few moments he reenters and declares, "Still not enough!" The bartender smiles and says, "My youngest just loves strawberry ice cream." How old are the children? A variant of the problem is for the sum of the ages to be 13 and the product of the ages to be the number posted over the door. In this case, it is the oldest that loves ice cream. Then how old are they? ==> logic/children.s <== First, determine all the ways that three ages can multiply together to get 72: 72 1 1 (quite a feat for the bartender) 36 2 1 24 3 1 18 4 1 18 2 2 12 6 1 12 3 2 9 4 2 9 8 1 8 3 3 6 6 2 6 4 3 As the man says, that's not enough information; there are many possibilities. So the bartender tells him where to find the sum of the ages--the man now knows the sum even though we don't. Yet he still insists that there isn't enough info. This must mean that there are two permutations with the same sum; otherwise the man could have easily deduced the ages. The only pair of permutations with the same sum are 8 3 3 and 6 6 2, which both add up to 14 (the bar's address). Now the bartender mentions his "youngest"--telling us that there is one child who is younger than the other two. This is impossible with 8 3 3--there are two 3 year olds. Therefore the ages of the children are 6, 6, and 2. Pedants have objected that the problem is insoluble because there could be a youngest between two three year olds (even twins are not born exactly at the same time). However, the word "age" is frequently used to denote the number of years since birth. For example, I am the same age as my wife, even though technically she is a few months older than I am. And using the word "youngest" to mean "of lesser age" is also in keeping with common parlance. So I think the solution is fine as stated. In the sum-13 variant, the possibilities are: 11 1 1 10 2 1 9 3 1 9 2 2 8 4 1 8 3 2 7 5 1 7 4 2 7 3 3 6 6 1 6 5 2 6 4 3 The two that remain are 9 2 2 and 6 6 1 (both products equal 36). The final bit of info (oldest child) indicates that there is only one child with the highest age. This cancels out the 6 6 1 combination, leaving the childern with ages of 9, 2, and 2. ==> logic/condoms.p <== How can a man have mutually safe sex with three women with only two condoms? How can three men have mutually safe sex with one woman with two condoms? ==> logic/condoms.s <== Use both condoms on the first woman. Take off the outer condom (turning it inside-out in the process) and set it aside. Use the inner condom alone on the second woman. Put the outer condom back on. Use it on the third woman. First man uses both condoms. Take off the outer condom (do NOT reverse it) and have second man use it. First man takes off the inner condom (turning it inside-out). Third man puts on this condom, followed by second man's condom. ==> logic/dell.p <== How can I solve logic puzzles (e.g., as published by Dell) automatically? ==> logic/dell.s <== #include <setjmp.h> #define EITHER if (S[1] = S[0], ! setjmp((S++)->jb)) { #define OR } else EITHER #define REJECT longjmp((--S)->jb, 1) #define END_EITHER } else REJECT; /* values in tmat: */ #define T_UNK 0 #define T_YES 1 #define T_NO 2 #define Val(t1,t2) (S->tmat[t1][t2]) #define CLASS(x) \ (((x) / NUM_ITEM) * NUM_ITEM) #define EVERY_TOKEN(x) \ (x = 0; x < TOT_TOKEN; x++) #define EVERY_ITEM(x, class) \ (x = CLASS(class); x < CLASS(class) + NUM_ITEM; x++) #define BEGIN \ struct state { \ char tmat[TOT_TOKEN][TOT_TOKEN]; \ jmp_buf jb; \ } States[100], *S = States; \ \ main() \ { \ int token; \ \ for EVERY_TOKEN(token) \ yes(token, token); \ EITHER /* Here is the problem-specific data */ #define NUM_ITEM 5 #define NUM_CLASS 6 #define TOT_TOKEN (NUM_ITEM * NUM_CLASS) #define HOUSE_0 0 #define HOUSE_1 1 #define HOUSE_2 2 #define HOUSE_3 3 #define HOUSE_4 4 #define ENGLISH 5 #define SPANISH 6 #define NORWEG 7 #define UKRAIN 8 #define JAPAN 9 #define GREEN 10 #define RED 11 #define IVORY 12 #define YELLOW 13 #define BLUE 14 #define COFFEE 15 #define TEA 16 #define MILK 17 #define OJUICE 18 #define WATER 19 #define DOG 20 #define SNAIL 21 #define FOX 22 #define HORSE 23 #define ZEBRA 24 #define OGOLD 25 #define PLAYER 26 #define CHESTER 27 #define LSTRIKE 28 #define PARLIA 29 char *names[] = { "HOUSE_0", "HOUSE_1", "HOUSE_2", "HOUSE_3", "HOUSE_4", "ENGLISH", "SPANISH", "NORWEG", "UKRAIN", "JAPAN", "GREEN", "RED", "IVORY", "YELLOW", "BLUE", "COFFEE", "TEA", "MILK", "OJUICE", "WATER", "DOG", "SNAIL", "FOX", "HORSE", "ZEBRA", "OGOLD", "PLAYER", "CHESTER", "LSTRIKE", "PARLIA", }; BEGIN yes(ENGLISH, RED); /* Clue 1 */ yes(SPANISH, DOG); /* Clue 2 */ yes(COFFEE, GREEN); /* Clue 3 */ yes(UKRAIN, TEA); /* Clue 4 */ EITHER /* Clue 5 */ yes(IVORY, HOUSE_0); yes(GREEN, HOUSE_1); OR yes(IVORY, HOUSE_1); yes(GREEN, HOUSE_2); OR yes(IVORY, HOUSE_2); yes(GREEN, HOUSE_3); OR yes(IVORY, HOUSE_3); yes(GREEN, HOUSE_4); END_EITHER yes(OGOLD, SNAIL); /* Clue 6 */ yes(PLAYER, YELLOW); /* Clue 7 */ yes(MILK, HOUSE_2); /* Clue 8 */ yes(NORWEG, HOUSE_0); /* Clue 9 */ EITHER /* Clue 10 */ yes(CHESTER, HOUSE_0); yes(FOX, HOUSE_1); OR yes(CHESTER, HOUSE_4); yes(FOX, HOUSE_3); OR yes(CHESTER, HOUSE_1); EITHER yes(FOX, HOUSE_0); OR yes(FOX, HOUSE_2); END_EITHER OR yes(CHESTER, HOUSE_2); EITHER yes(FOX, HOUSE_1); OR yes(FOX, HOUSE_3); END_EITHER OR yes(CHESTER, HOUSE_3); EITHER yes(FOX, HOUSE_2); OR yes(FOX, HOUSE_4); END_EITHER END_EITHER EITHER /* Clue 11 */ yes(PLAYER, HOUSE_0); yes(HORSE, HOUSE_1); OR yes(PLAYER, HOUSE_4); yes(HORSE, HOUSE_3); OR yes(PLAYER, HOUSE_1); EITHER yes(HORSE, HOUSE_0); OR yes(HORSE, HOUSE_2); END_EITHER OR yes(PLAYER, HOUSE_2); EITHER yes(HORSE, HOUSE_1); OR yes(HORSE, HOUSE_3); END_EITHER OR yes(PLAYER, HOUSE_3); EITHER yes(HORSE, HOUSE_2); OR yes(HORSE, HOUSE_4); END_EITHER END_EITHER yes(LSTRIKE, OJUICE); /* Clue 12 */ yes(JAPAN, PARLIA); /* Clue 13 */ EITHER /* Clue 14 */ yes(NORWEG, HOUSE_0); yes(BLUE, HOUSE_1); OR yes(NORWEG, HOUSE_4); yes(BLUE, HOUSE_3); OR yes(NORWEG, HOUSE_1); EITHER yes(BLUE, HOUSE_0); OR yes(BLUE, HOUSE_2); END_EITHER OR yes(NORWEG, HOUSE_2); EITHER yes(BLUE, HOUSE_1); OR yes(BLUE, HOUSE_3); END_EITHER OR yes(NORWEG, HOUSE_3); EITHER yes(BLUE, HOUSE_2); OR yes(BLUE, HOUSE_4); END_EITHER END_EITHER /* End of problem-specific data */ solveit(); OR printf("All solutions found\n"); exit(0); END_EITHER } no(a1, a2) { int non1, non2, token; if (Val(a1, a2) == T_YES) REJECT; else if (Val(a1, a2) == T_UNK) { Val(a1, a2) = T_NO; no(a2, a1); non1 = non2 = -1; for EVERY_ITEM(token, a1) if (Val(token, a2) != T_NO) if (non1 == -1) non1 = token; else break; if (non1 == -1) REJECT; else if (token == CLASS(a1) + NUM_ITEM) yes(non1, a2); for EVERY_TOKEN(token) if (Val(token, a1) == T_YES) no(a2, token); } } yes(a1, a2) { int token; if (Val(a1, a2) == T_NO) REJECT; else if (Val(a1, a2) == T_UNK) { Val(a1, a2) = T_YES; yes(a2, a1); for EVERY_ITEM(token, a1) if (token != a1) no(token, a2); for EVERY_TOKEN(token) if (Val(token, a1) == T_YES) yes(a2, token); else if (Val(token, a1) == T_NO) no(a2, token); } } solveit() { int token, tok2; for EVERY_TOKEN(token) for (tok2 = token; tok2 < TOT_TOKEN; tok2++) if (Val(token, tok2) == T_UNK) { EITHER yes(token, tok2); OR no(token, tok2); END_EITHER; return solveit(); } printf("Solution:\n"); for EVERY_ITEM(token, 0) { for (tok2 = NUM_ITEM; tok2 < TOT_TOKEN; tok2++) if (Val(token, tok2) == T_YES) printf("\t%s %s\n",names[token],names[tok2]); printf("\n"); } REJECT; } --- james@crc.ricoh.com (James Allen) ==> logic/elimination.p <== 97 baseball teams participate in an annual state tournament. The way the champion is chosen for this tournament is by the same old elimination schedule. That is, the 97 teams are to be divided into pairs, and the two teams of each pair play against each other. After a team is eliminated from each pair, the winners would be again divided into pairs, etc. How many games must be played to determine a champion? ==> logic/elimination.s <== In order to determine a winner all but one team must lose. Therefore there must be at least 96 games. ==> logic/flip.p <== How can a toss be called over the phone (without requiring trust)? ==> logic/flip.s <== A flips a coin. If the result is heads, A multiplies 2 prime numbers containing about 90 digits; if the result is tails, A multiplies 3 prime numbers containing about 60 digits. A tells B the result of the multiplication. B now calls either heads or tails and tells A. A then supplies B with the original numbers to verify the flip. Consider what is involved in multiplying 90 digit numbers. Using the method of long multiplication that we all learned in grade school, you have maybe 90 or so strings of 90 characters (or "digits") each. That's no problem for a computer to deal with. The magnitude of the number represented isn't much of a factor; we're only manipulating the string of digits. Consider what is involved in factoring 90 digit numbers. There are of course, little tricks for determining factorability by small integers which we all learned in grade school. (Is the last digit even? Is the sum of all the digits divisible by 9? And so on.) But these are of little use in factoring large numbers with large factors. In fact, there is no essentially better method than checking every number smaller that the number to be factored and seeing if it one divides the other evenly. We means we could be checking some 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 nummbers. This is very hard to do, even for a supercomputer. Here, the number of digits this number has is of little consequence. It is the magnitude of the number that we have to worry about, and there just isn't enough time in the world to do this properly. Where does A find a list of 60- and 90- digit prime numbers? Well, that's not to hard to come by. The simplest method to find a few prime numbers is simply to choose a 90-digit number (or 60-digit number, as the case warrants) randomly, and check to see if it is prime. You won't have to wait too long before you stumble across a prime number. "But wait!" you cry. "I thought you just said it was incredibly difficult to factor large numbers. If that's the case, then how are you going to know if the number you randomly choose is prime?" A good question. Here we enter into the strange an wacky world of number theory. It turns out (and I won't explain how unless someone asks) there are "probabalistic" algorithms, which depend on us choosing numbers at random. There are probablistic algorithms that when given a prime number, will quickly tell us that it is a prime number, and when given a composite number, will either tell us that it is a composite number (with very, very high probability) or will tell us that it is a prime number (with very, very low probability.) What's the use of an algorithm that only returns the right answer "with very, very high probability?" Well, we can make this probability as high as we want, just by running the algorithm longer. In fact, it is within our technological abilities to quickly run this algorithm for 90-digit numbers so that the probability of it giving a wrong answer is less than the probability of a cosmic ray striking a vital part of the computer at some vital time and causing the computer to spit out the wrong answer anyway. That's what I mean by "very, very high." ==> logic/flowers.p <== How many flowers do I have if all of them are roses except two, all of them are tulips except two, and all of them are daisies except two? ==> logic/flowers.s <== There are two solutions: Three flowers: rose, tulip, daisy Two flowers: carnation, geranium ==> logic/friends.p <== Any group of 6 or more contains either 3 mutual friends or 3 mutual strangers. Prove it. ==> logic/friends.s <== Take a person X. Of the five other people, there must either be at least three acquaintances of X or at least three strangers of X. Assume wlog that X has three strangers A,B,C. Unless A,B,C is the required triad of acquaintances, they must include a pair of strangers, wlog A,B. Then X,A,B is the required triad of strangers, QED. ==> logic/hofstadter.p <== In first-order logic, find a predicate P(x) which means "x is a power of 10." ==> logic/hofstadter.s <== Well, one summer, I decided to tackle the problem. Not having any great knowledge of number theory, I used a more brute force approach. Note that for greater comprehensibility, I have broken the resulting formula up into several stages, but it would not be difficult to put it back together into one vast formula: {e is prime:} PRIME(e) := ~Eb:Ec:((b+2)*(c+2) = e) {if e is a prime, true iff a is a power of e:} PPOWER(a,e) := Ab:Ac:((b*c = a) -> ((b = 1) or (Ed: (e*d) = b))) {if e is a prime, and a is a power of e, true iff d is the (log_e a)th digit (counting from the right, starting with 0) in the base-e expansion of n:} DIG(a,e,d,n) := (d < e) & Eb:Ec:((c < a) & (n = (b*e*a) + (d*a) + c)) {if e is prime, and a is a power of e, true iff n has exactly (log_e a) digits in its base-e expansion (0 is counted as having 0 digits:} LENGTH(e,a,n):= (n < a) & Ab:((PPOWER(b,e) & (b < a)) -> (b <= n)) POWER_OF_TEN(x):= Ee:(PRIME(e) & (e > x) & En:Ea:(LENGTH(e,a,n) & DIG(1,e,1,n) & Ai:Aj:Au:( (PPOWER(u,e) & ((e*u) < a) & DIG(u,e,i,n) & DIG(e*u,e,j,n)) -> j = (10 * i) ) & Eu:(PPOWER(u,e) & (e*u = a) & DIG(u,e,x,n)) ) ) The basic idea is that you are asserting that for some prime e greater than x, we can find a number n which, when expressed in base-e notation, will have 1 in its units place, 10 in its e's place, 100 in its (e^2)'s place, and in general have the "digit" in each place be 10 times greater than the one to its right, such that the leftmost digit is our x. To translate into Hofstadter's notation, of course, we must use horse-shoes instead of ->'s, big carets instead of &'s, letters a through e (followed by however many ''s) exclusively, and so forth. (We must also replace <'s and <= with appropriate expansions, and substitute in for our capitalized formula abbreviations.) This is left as an exercise to the reader. Kevin Wald wald2@husc.harvard.edu ==> logic/hundred.p <== A sheet of paper has statements numbered from 1 to 100. Statement n says "exactly n of the statements on this sheet are false." Which statements are true and which are false? What if we replace "exactly" by "at least"? ==> logic/hundred.s <== From _Litton's Problematical Recreations_ It is tempting to argue as follows: At most one statement can be true (they are mutually contradictory). If they are all false, statement 100 would be true, which is no good. So 99 are false, and statement 99 is true. If replaced by "at least", and the "real" number of false statements is x, then statements x+1 to 100 will be false (since they falsely claim that there are more false statements than there actually are). So, 100-x are false, ie. x=100-x, so x=50. The first 50 statements are true, and statements 51 to 100 are false. However, there is a hidden and incorrect assumption in this argument. To see this, suppose that there is one statement on the sheet and it says "One statement is false" or "At least one statement is false," either way it implies "this statement is false," which is a familiar paradoxical statement. We have learned that this paradox arises because of the false assumption that all statements are either true or false. This is the hidden assumption in the above reasoning. If it is acknowledged that some of the statements on the page may be neither true nor false (i.e., meaningless), then nothing whatsoever can be concluded about which statements are true or false. This problem has been carefully contrived to appear to be solvable (like the vacuous statement "this statement is true"). By changing the numbers in some statements and changing "true" to "false," various circular forms of the liar's paradox can be constructed. A much more complicated version of the same problem is: 1) At least one of the last two statements in this list is true. 2) This is either the first true or the first false statement in the list. 3) There exist three consecutive false statements. 4) The difference beween the numbers of the last true statement and the first true statement is a factor of the unknown number. 5) The sum of the numbers of the true statements is the unknown number. 6) This is not the last true statement. 7) Each true statement's number is a factor of the unknown number. 8) The unknown number equals the percentage of these statements which are true. 9) The number of different factors which the unknown number has (excluding 1 and itself) is more than the sum of the numbers of the true statements. 10) There are no three cosecutive true statements. What is the number? The incorrect but plausible solution is: By 2, either way 1 must be false, and then so must both 9 and 10. 6, if false, says "This is the last true statement", which gives a paradox, thus 6 must be true, and so must 7 and/or 8. 7 and 8 cannot both be true, as the number had to be a multiple of 6,7,8 , that is a multiple of 168 (by 7), and less than 100 (by 8) If 8 is false, then 3 is true (8,9,10 is false), if 8 is true, then 3 is false (3 cannot be true when both 6 and 8 are true, then there are no three consecutive statements left). So we have either A) F X F X X T F T F F or B) F X T X X T T F F F In A), 4 and 5 must be true (by false 10), and 2 may be true or false. So by 5 the number shall be either 27 (2+3+4+5+6+7) or 25 (3+4+5+6+7). None of these can fullfill 8, though, so A) is out leaving us with B) Now, (by 7) 3,6 and 7 are factors, the number must be a multiple of 42, as 2+3+4+5+6+7=27, 5 must be false. By false 10, 2 and 4 must be true, that is 5 shall be a multiple of the number. Now the number must be a multiple of 2,3,4,5,6 and 7, that is a multiple of 3*4*5*7=420. 420 has 22 different factors (2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210) and the sum 2+3+4+6+7 = 22, so the only multiple of 420 that fulfills false 9 is 420. ==> logic/inverter.p <== Can a digital logic circuit with two inverters invert N independent inputs? The circuit may contain any number of AND or OR gates. ==> logic/inverter.s <== It can be shown that N inverters can invert 2^N-1 independent inputs, given an unlimited supply of AND and OR gates. The classic version of this puzzle is to invert 3 independent inputs using AND gates, OR gates, and only 2 inverters. This is solved by: n1 = not(i1 and i2 or i1 and i3 or i2 and i3); n2 = not((i1 or i2 or i3) and n1 or i1 and i2 and i3); o1 = (i2 or i3 or n2) and n1 or i2 and i3 and n2; o2 = (i1 or i3 or n2) and n1 or i1 and i3 and n2; o3 = (i1 or i2 or n2) and n1 or i1 and i2 and n2; i1, i2, and i3 are the inputs, n1 and n2 are the inverted signals, and o1, o2, and o3 are the outputs. "and" has higher precedence than "or". So, start with N inverters. Replace 3 of them with 2. Keep doing that until you're down to 2 inverters. I was skeptical at first, because such a design requires so much feedback that I was sure the system would oscillate when switching between two particular states. But after writing a program to test every possible state change (32^2), it appears that this system settles after a maximum of 3 feedback logic iterations. I did not include gate delays in the simulation, however, which could increase the number of iterations before the system settles. In any case, it appears that the world needs only 2 inverters! :-) ==> logic/josephine.p <== The recent expedition to the lost city of Atlantis discovered scrolls attributted to the great poet, scholar, philosopher Josephine. They number eight in all, and here is the first. THE KINGDOM OF MAMAJORCA, WAS RULED BY QUEEN HENRIETTA I. IN MAMAJORCA WOMEN HAVE TO PASS AN EXTENSIVE LOGIC EXAM BEFORE THEY ARE ALLOWED TO GET MARRIED. QUEENS DO NOT HAVE TO TAKE THIS EXAM. ALL THE WOMEN IN MAMAJORCA ARE LOYAL TO THEIR QUEEN AND DO WHATEVER SHE TELLS THEM TO. THE QUEENS OF MAMAJORCA ARE TRUTHFUL. ALL SHOTS FIRED IN MAMAJORCA CAN BE HEARD IN EVERY HOUSE. ALL ABOVE FACTS ARE KNOWN TO BE COMMON KNOWLEDGE. HENRIETTA WAS WORRIED ABOUT THE INFIDELITY OF THE MARRIED MEN IN MAMAJORCA. SHE SUMMONED ALL THE WIVES TO THE TOWN SQUARE, AND MADE THE FOLLOWING ANNOUNCEMENT. "THERE IS AT LEAST ONE UNFAITHFUL HUSBAND IN MAMAJORCA. ALL WIVES KNOW WHICH HUSBANDS ARE UNFAITHFUL, BUT HAVE NO KNOWLEDGE ABOUT THE FIDELITY OF THEIR OWN HUSBAND. YOU ARE FORBIDDEN TO DISCUSS YOUR HUSBAND'S FAITHFULNESS WITH ANY OTHER WOMAN. IF YOU DISCOVER THAT YOUR HUSBAND IS UNFAITHFUL, YOU MUST SHOOT HIM AT PRECISELY MIDNIGHT OF THE DAY YOU FIND THAT OUT." THIRTY-NINE SILENT NIGHTS FOLLOWED THE QUEEN'S ANNOUNCEMENT. ON THE FORTIETH NIGHT, SHOTS WERE HEARD. QUEEN HENRIETTA I IS REVERED IN MAMAJORCAN HISTORY. As with all philosophers Josephine doesn't provide the question, but leaves it implicit in his document. So figure out the questions - there are two - and answer them. Here is Josephine's second scroll. QUEEN HENRIETTA I WAS SUCCEEDED BY DAUGHTER QUEEN HENRIETTA II. AFTER A WHILE HENRIETTA LIKE HER FAMOUS MOTHER BECAME WORRIED ABOUT THE INFIDELITY PROBLEM. SHE DECIDED TO ACT, AND SENT A LETTER TO HER SUBJECTS (WIVES) THAT CONTAINED THE EXACT WORDS OF HENRIETTA I'S FAMOUS SPEECH. SHE ADDED THAT THE LETTERS WERE GUARENTEED TO REACH ALL WIVES EVENTUALLY. QUEEN HENRIETTA II IS REMEMBERED AS A FOOLISH AND UNJUST QUEEN. What is the question and answer implied by this scroll? ==> logic/josephine.s <== The two questions for scroll #1 were: 1. How many husbands were shot on that fateful night? 2. Why is Queen Henrietta I revered in Mamajorca? The answers are: If there are n unfaithful husbands (UHs), every wife of an UH knows of n-1 UH's while every wife of a faithful husband knows of n UHs. [this because everyone has perfect information about everything except the fidelity of their own husband]. Now we do a simple induction: Assume that there is only one UH. Then all the wives but one know that there is just one UH, but the wife of the UH thinks that everyone is faithful. Upon hearing that "there is at least one UH", the wife realizes that the only husband it can be is her own, and so shoots him. Now, imagine that there are just two UH's. Each wife of an UH assumes that the situation is "only one UH in town" and so waits to hear the other wife (she knows who it is, of course) shoot her husband on the first night. When no one is shot, that can only be because her OWN husband was a second UH. The wife of the second UH makes the same deduction when no shot is fired the first night (she was waiting, and expecting the other to shoot, too). So they both figure it out after the first night, and shoot their husbands the second night. It is easy to tidy up the induction to show that the n UHs will all be shot just on the n'th midnight. The question for scroll #2 is: 3. Why is Queen Henrietta II not? The answer is: The problem now is that QHII didn't realize that it is *critical* that all of the wives, of faithful and UH's alike, to *BEGIN*AT*THE*SAME*MOMENT*. The uncertainty of having a particular wife's notice come a day or two late makes the whole logic path fall apart. That's why she's foolish. She is unjust, because some wives, honed and crack logicians all, remember, will *incorrectly* shoot faithful husbands. Let us imagine the situation with just a SINGLE UH in the whole country. And, wouldn't you know it, the notice to the wife of the UH just happens to be held up a day, whereas everyone else's arrived the first day. Now, all of the wives that got the notice the first day know that there is just one UH in the country. And they know that the wife of that UH will think that everyone is faithful, and so they'll expect her to figure it out and shoot her husband the first night. BUT SHE DIDN"T GET THE NOTICE THE FIRST NIGHT.... BUT THE OTHER WIVES HAVE NO WAY OF KNOWING THAT. So, the wife of the UH doesn't know that anything is going on and so (of course) doesn't do anything the first night. The next day she gets the notice, figures it all out, and her husband will be history come that midnight. BUT... *every* other wife thought that there should have been a shooting the first night, and since there wasn't there must have been an additional UH, and it can only have been _her_ husband. So on the second night **ALL** of the husbands are shot. Things are much more complicated if the mix of who gets the notice when is less simple than the one I mentioned above, but it is always wrong and/or tragic. NOTE: if the wives *know* that the country courier service (or however these things get delivered) is flaky, then they can avoid the massacre, but unless the wives exchange notes no one will ever be shot (since there is always a chance that rather than _your_ husband being an UH, you could reason that it might be that the wife of one of the UH's that you know about just hasn't gotten her copy of the scroll yet). I guess you could call this case "unjust", too, since the UH's evade punishment, despite the perfect logic of the wives. ==> logic/locks.and.boxes.p <== You want to send a valuable object to a friend. You have a box which is more than large enough to contain the object. You have several locks with keys. The box has a locking ring which is more than large enough to have a lock attached. But your friend does not have the key to any lock that you have. How do you do it? Note that you cannot send a key in an unlocked box, since it might be copied. ==> logic/locks.and.boxes.s <== Attach a lock to the ring. Send it to her. She attaches her own lock and sends it back. You remove your lock and send it back to her. She removes her lock. ==> logic/min.max.p <== In a rectangular array of people, which will be taller, the tallest of the shortest people in each column, or the shortest of the tallest people in each row? ==> logic/min.max.s <== Let T denote shortest of tall Let S denote tallest of short ------------------------------- | | | | | S | | | | | | T X | | | | | | | | | ------------------------------- So T >= X >= S. ==> logic/mixing.p <== Start with a half cup of tea and a half cup of coffee. Take one tablespoon of the tea and mix it in with the coffee. Take one tablespoon of this mixture and mix it back in with the tea. Which of the two cups contains more of its original contents? ==> logic/mixing.s <== Mixing Liquids The two cups end up with the same volume of liquid they started with. The same amount of tea was moved to the coffee cup as coffee to the teacup. Therefore each cup contains the same amount of its original contents. ==> logic/monty.52.p <== Monty and Waldo play a game with N closed boxes. Monty hides a dollar in one box; the others are empty. Monty opens the empty boxes one by one. When there are only two boxes left Waldo opens either box; he wins if it contains the dollar. Prior to each of the N-2 box openings Waldo chooses one box and locks it, preventing Monty from opening it next. That box is then unlocked and cannot be so locked twice in a row. What are the optimal strategies for Monty and Waldo and what is the fair price for Waldo to pay to play the game? ==> logic/monty.52.s <== The fair price for large N is $0.6321205588285576784; I will offer a proof along with optimal strategies. Denote the game as G_N(). After (N-M) rounds of play, the game will have the same form as G_M(). Depending on the strategies each of the M boxes will have a probability p_i of containing the dollar. Let Waldo lock the M'th box (renumbering the boxes if necessary). Denote the game and Waldo's expected winnings in the game by G_M(p_1, p_2, ..., p_M) where p_1 + p_2 + ... + p_M = 1 When p_2 = p_3 = p_4 = ... = p_M we adopt the abbreviation G_M(b) = G_M(1 + b - Mb, b, b, b, b, ..., b) and note that since probabilities are never negative: 1 + b - Mb >= 0, or 0 <= b <= 1 / (M-1) Various G_M(p_1, p_2, ..., p_M) have difficult solutions but we are asked only to solve G_M(1/M) and it turns out we can accomplish this by considering only the games G_M(b) where 1/M <= b <= 1/(M-1) [1] Games of this form will be said to satisfy constraint [1]. Induction is used for one of the theorems, so we'd better solve G_2 and G_3 for our basis. G_2(p_1, p_2) = max (p_1, p_2) G_3(p_1, p_2, p_3) = max (p_1 + p_2, p_3) since after Monty opens box #1, box #2 will have probability (p_1 + p_2) and vice versa. When the probabilities satisfy constraint [1]: G_2(b) = G_2(1-b, b) = b G_3(b) = G_3(1-2b, b, b) = 1 - b The proof of Theorem 1 is based on the probability p_i that box #i contains the dollar. (Of course this is Waldo's perceived probability: Monty's probability would be 0 or 1.) It may seem wrong for Waldo to "forget" the game history and remember only the computed p_i. For example he may have previously locked some but not all of the boxes and this fact is ignored except in the calculation of p_i. Or Monty may have some higher level "plan" which mightn't seem to translate directly into probabilities. But probability algebra obeys some simplifying linearity rules and, if Monty keeps a "poker face", the probability model is the only thing Waldo has to act on. Especially paradoxical is the derivation of Waldo's p_i in his trivial strategy below: he can adopt inferior but "correct" p_i to simplify the analysis. Theorem 1) If b >= 1/M then G_M(b) = G_[M-1]( (1-b) / (M-2) ) Proof) We will show that Monty and Waldo each have a strategy in G_M(b) to reduce the game to G_[M-1](b, q, q, ..., q) where q = (1-b) / (M-2) and where the boxes have been renumbered so that box #1 was box #M (the one Waldo locked) from the prior round and the new box #(M-1) is the one Waldo locks next. Note that if Monty indeed arranges the probability mixture G_[M-1](b, q, q, q, q, ..., q) it won't matter which box Waldo locks (Box #1 has the only non-equal probability but Waldo cannot lock the same box twice in a row); this is a typical property of "saddlepoint" strategy. We will derive the necessary and sufficient condition for Monty to reduce any game G_M(p_1, p_2, p_3, ..., p_M) to a single game with the form G_[M-1](b, q, q, ..., q). Using the numbering of G_M() let R(i,j) be the joint probability that Box #i contains the dollar and Box #j is opened by Monty in G_M(). We need consider only M >= 3 Clearly, R(i, j) >= 0 R(i, i) = 0 R(i, M) = 0, i < M sum_over_j R(i,j) = p_i and to achieve q_2 = q_3 = ... = q_[M-1] in G_[M-1], R(1, j) = R(k, j) for 1 < j,k < M and j != k R(2, 1) = R(k, 1) for 2 < k < M and to make G_[M-1] be independent of Monty's play R(M, j)/R(1, j) = R(M, 2)/R(1, 2) for 2 < j < M R(M, 2)/R(1, 2) = R(M, 1)/R(2, 1) The above have a simple unique solution: R(i, j) = (1 - p_M) / (M - 2) - p_j [2] for i,j < M and i != j R(M, j) = p_M - p_j * p_M / (1 - p_M) [3] for j < M p_j * (M-2) + p_M <= 1 [4] For the theorem we are given that G_M(b) satisfies constraint [1] 1 / M <= b <= 1 / (M - 1) which implies the weaker inequality (M - 3) / (M^2 - 3M + 1) <= b <= 1 / (M - 1) and since for the constraint-[1] compliant G_M() p_j = b or p_j = (1+b-Mb) for all j the inequality [4] follows directly. Hence Monty can transpose G_M(b) into G_[M-1]( (1-b) / (M-2) ) whenever b >= 1/M and M >= 3. (This G_[M-1] also happens to satisfy constraint [1] as needed for the next theorem.) It should be easy to argue that this strategy is optimal for Monty, but we want to derive Waldo's best strategy anyway and if it guarantees the same value we know we're at the "saddlepoint". If Waldo knows Monty has a non-optimal strategy he can take advantage of it, but we will just derive a strategy good enough to achieve the saddle-point value. Monty must transform G_M(b) into some G_[M-1](b, q_2, q_3, ..., q_[M-1]) where Waldo has the choice of locking any of boxes #2 through #(M-1). If Waldo locks each of the (M-2) available boxes with probability 1/(M-2) it is easily seen that the average probability that he locks the box with the dollar is (1-b) / (M-2) and the probabilities q_2, q_3, ..., q_[M-2] will also have the average value (1-b)/(M-2). If Waldo pretends to "forget" which box he picked before, he can take q_2 = q_3 = ... = (1-b)/(M-2) thereby constructing the same game Monty constructed with his saddlepoint strategy! In the above Waldo in effect "degraded" the accuracy of his probability estimates with the substitutions q_2' = (q_2 + q_3 + ... + q_[M-1]) / (M - 2) q_3' = (q_2 + q_3 + ... + q_[M-1]) / (M - 2) et cetera If Waldo "knows" more than this, he can pretend he doesn't! For example he can ask Monty to secretly shuffle the boxes. Thus either player can reduce G_M(b), b >= 1/M to G_[M-1]((1-b)/(M-2)) so this must be the saddlepoint. Q.E.D. Theorem 2) If b >= 1/M then G_M(b) = 1 - 1/2! + 1/3! - ... - (1-b)(-1)^M/(M-2)! = - sum (-1)^i/i! - (1-b)(-1)^M/(M-2)! where the sum is over i = 1, 2, 3, ..., M-3 Proof) The proof is by induction. We know the theorem holds for M = 3 and we will assume it holds for (M-1). Set c = (1-b) / (M-2) We noted earlier that b <= 1/(M-1): otherwise p_1 = (1 + b - Mb) is negative; hence we obtain c = (1-b)/(M-2) >= (1 - 1/(M-1)) / (M-2) or simply c >= 1/(M-1) Thus the condition of the inductive hypothesis is satisfied and G_[M-1](c) = 1 - 1/2! + 1/3! - ... + (1-c)(-1)^M/(M-3)! But from Theorem 1 G_M(b) = G_[M-1](c) and from the definition of c, c/(M-3)! = (1-b)/(M-2)! which establishes the theorem. Theorem 3) G_M(1/M) = G_M(1/M, ..., 1/M) = 1 - 1/2! + 1/3! - ... -(-1)^M/M! Proof) This follows directly from Theorem 2 and the observation that (1/M)/(M-2)! = 1/(M-1)! - 1/M! For large M, G_M(1/M) approaches (1 - 1/e). It will be a little bigger when M is odd and a little smaller when M is even. I've appended the numeric values below. % dc [[Solution for M =]Plb1+pdsb]sy 65k1sa1sblyx2sc[la1lc/-dsaplclyx*scla1lc/+dsaplclyx*sclzx]dszx Solution for M =2 0.50000000000000000000000000000000000000000000000000000000000000000 Solution for M =3 0.66666666666666666666666666666666666666666666666666666666666666666 Solution for M =4 0.62500000000000000000000000000000000000000000000000000000000000000 Solution for M =5 0.63333333333333333333333333333333333333333333333333333333333333333 Solution for M =6 0.63194444444444444444444444444444444444444444444444444444444444445 Solution for M =7 0.63214285714285714285714285714285714285714285714285714285714285714 Solution for M =8 0.63211805555555555555555555555555555555555555555555555555555555556 Solution for M =9 0.63212081128747795414462081128747795414462081128747795414462081129 Solution for M =10 0.63212053571428571428571428571428571428571428571428571428571428572 . . . Solution for M =52 0.63212055882855767840447622983853913255418886896823216549216319831 P. S. There are related unsolved problems: (a) what about G_M(p_1, p_2, ..., p_M) that do not fit the pattern used in the above solution? (b) what if two boxes contain dollars? (first, what should the rules be?) -- james@crc.ricoh.com (James Allen) ==> logic/number.p <== Mr. S. and Mr. P. are both perfect logicians, being able to correctly deduce any truth from any set of axioms. Two integers (not necessarily unique) are somehow chosen such that each is within some specified range. Mr. S. is given the sum of these two integers; Mr. P. is given the product of these two integers. After receiving these numbers, the two logicians do not have any communication at all except the following dialogue: <<1>> Mr. P.: I do not know the two numbers. <<2>> Mr. S.: I knew that you didn't know the two numbers. <<3>> Mr. P.: Now I know the two numbers. <<4>> Mr. S.: Now I know the two numbers. Given that the above statements are absolutely truthful, what are the two numbers? ==> logic/number.s <== The answer depends upon the ranges from which the numbers are chosen. The unique solution for the ranges [2,62] through [2,500+] is: SUM PRODUCT X Y 17 52 4 13 The unique solution for the ranges [3,94] through [3,500+] is: SUM PRODUCT X Y 29 208 13 16 There are no unique solutions for the ranges starting with 1, and there are no solutions for ranges starting with numbers above 3. A program to compute the possible pairs is included below. #include <stdio.h> /* BEGINNING OF PROBLEM STATEMENT: Mr. S. and Mr. P. are both perfect logicians, being able to correctly deduce any truth from any set of axioms. Two integers (not necessarily unique) are somehow chosen such that each is within some specified range. Mr. S. is given the sum of these two integers; Mr. P. is given the product of these two integers. After receiving these numbers, the two logicians do not have any communication at all except the following dialogue: <<1>> Mr. P.: I do not know the two numbers. <<2>> Mr. S.: I knew that you didn't know the two numbers. <<3>> Mr. P.: Now I know the two numbers. <<4>> Mr. S.: Now I know the two numbers. Given that the above statements are absolutely truthful, what are the two numbers? END OF PROBLEM STATEMENT */ #define SMALLEST_MIN 1 #define LARGEST_MIN 10 #define SMALLEST_MAX 50 #define LARGEST_MAX 500 long P[(LARGEST_MAX + 1) * (LARGEST_MAX + 1)]; /* products */ long S[(LARGEST_MAX + 1) + (LARGEST_MAX + 1)]; /* sums */ find(long min, long max) { long i, j; /* * count factorizations in P[] * all P[n] > 1 satisfy <<1>>. */ for(i = 0; i <= max * max; ++i) P[i] = 0; for(i = min; i <= max; ++i) for(j = i; j <= max; ++j) ++P[i * j]; /* * decompose possible SUMs and check factorizations * all S[n] == min - 1 satisfy <<2>>. */ for(i = min + min; i <= max + max; ++i) { for(j = i / 2; j >= min; --j) if(P[j * (i - j)] < 2) break; S[i] = j; } /* * decompose SUMs which satisfy <<2>> and see which products * they produce. All (P[n] / 1000 == 1) satisfy <<3>>. */ for(i = min + min; i <= max + max; ++i) if(S[i] == min - 1) for(j = i / 2; j >= min; --j) if(P[j * (i - j)] > 1) P[j * (i - j)] += 1000; /* * decompose SUMs which satisfy <<2>> again and see which products * satisfy <<3>>. Any (S[n] == 999 + min) satisfies <<4>> */ for(i = min + min; i <= max + max; ++i) if(S[i] == min - 1) for(j = i / 2; j >= min; --j) if(P[j * (i - j)] / 1000 == 1) S[i] += 1000; /* * find the answer(s) and print them */ printf("[%d,%d]\n",min,max); for(i = min + min; i <= max + max; ++i) if(S[i] == 999 + min) for(j = i / 2; j >= min; --j) if(P[j * (i - j)] / 1000 == 1) printf("{ %d %d }: S = %d, P = %d\n", i - j, j, i, (i - j) * j); } main() { long min, max; for (min = SMALLEST_MIN; min <= LARGEST_MIN; min ++) for (max = SMALLEST_MAX; max <= LARGEST_MAX; max++) find(min,max); } ------------------------------------------------------------------------- = Jeff Kenton (617) 894-4508 = = jkenton@world.std.com = ------------------------------------------------------------------------- ## User Contributions:## Comment about this article, ask questions, or add new information about this topic: |

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