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Problem 1
A tennis club invites 32 (2⁵) players of equal ability to compete in an elimination tournament. If both John and Jim Smith are invited, what is the chance of their playing each other during the tournament?

Problem 2
What are the last three digits of the number 7⁹⁹⁹⁹?

Problem 3
Three men play a game with the understanding that the loser is to double the money of the other two. After three games, each has lost just once; and each has $24. How much did each have to start?

Problem 4
A Mathematician whose clock had stopped wound it, but did not bother to set it correctly. Then he walked from his home to the home of a friend for an evening of music. Afterwards, he walked back to his own home and set his clock exactly. How could he do this without knowing the time his trip took?

Problem 5
A wizard in numerical analysis has a gold chain with 7 links. A lady programmer challenges him to use the chain to buy 7 kisses, each kiss to be paid for, separately, with one chain link. What is the smallest number of cuts he will have to make in the chain? What is his sequence of payments?

Problem 6
A man started for a walk when the hands of his watch were coincident between three and four o'clock. When he finished, the hands were again coincident between five and six o'clock. What was the time when he started, and how long did he walk?

Problem 7
A pet store offered a baby monkey for sale at $1.25. The monkey grew. Next week it was offered at $1.89, then $5.13, then $5.94, then $9.18 and on the sixth week a Ph.D. in Aeronautics bought it for $12.42. How were the new prices figured?

Problem 8
Two candles have equal lengths. One is consumed uniformly in four hours, the other in five hours. If they are lighted at the same time, when will one be three times as long as the other?

Problem 9
A ball is dropped from a height of 10 feet. It rebounds one-half the distance on each bounce. What is the total distance it travels?

Problem 10
A college president, a professor, an instructor, and a janitor are named Mr. Brown, Mr. Green, Mr. White, and Mr. Black, but not respectively. Four students with the same names will be designated here as Brow, Green, White and Black. The student with the same name as the professor belongs to Black's fraternity. Mr. Green's daughter-in-law lives in Philadelphia. The father of one of the students always confuses White with Green in class, but is not absent-minded. The janitor's wife has never seen Mr. Black. Mr. White is the instructor's father-in-law and has no grandchildren. The president's oldest son is seven. What are the names of the president, professor, instructor, and janitor?

Problem 11
A hunter wished to take his one-piece rifle on a train but the conductor refused to permit it in the coach and the baggage man could not take any article whose greatest dimension exceeded 1 yard. The length of the rifle was 1.7 yards. What could the hunter do?

Problem 12
In how many zeros does 10,000! (factorial) end?

Problem 13
In a contest: Jim beat Frank and John; Frank beat Joe, Tom, and John; Joe beat Jim and Tom; Tom beat Jim and John; and John beat Joe. Rank the players according to their winning ability.

Problem 14
Five cards are drawn at random from a pack of cards which have been numbered consecutively from 1 to 97, and thoroughly shuffled. What is the probability that the numbers on the cards as drawn are in increasing order of magnitude?

Problem 15
Four snails start at the vertices of a unit square and move directly toward one another in cyclic order, at unit rate. How far will they travel before they meet?

Problem 16
If each of the letters A, B, and C represents a particular numeric digit, what is the minimum value of the whole number ABC divided by A + B + C? Hint: the answer is not 1.

Problem 17
Around a cylindrical tub, outside circumference 4", length 9", 10 turns of a wire are helically wound. The ends of the wire coincide with the ends of the same cylindrical element. Find the length of the wire.

Problem 18
What number, if divided by 10, leaves a remainder of 9; divided by 9 leaves a remainder of 8; divided by 8 leaves a remainder of 7, …, divided by 2 leaves a remainder of 1. One answer is 14,622,042,959. Find a smaller solution.

Problem 19
Find the smallest number with 28 divisors.

Problem 20
Two flights of bombers were flying at 300 mph on converging courses 30 degrees apart, each flight being 240 miles from the rendezvous. From above each flight a fighter plan, flying at 500 mph, flew to the other bomber flight and returned, continuing the shuttle until the bomber flights met. One fighter always headed directly toward his objective, while the other fighter always flew an interception course. Which fighter flew the greater distance, and how much farther did he fly?

Problem 21
If P_n denotes the nth prime, show that P₁P₂…P_n + 1 is not a perfect square.

Problem 22
For X < 1 evaluate the infinite product: (1 + X + X² + … + X₉)(1 + X¹⁰ + X²⁰ + X³⁰ + … + X⁹⁰)(1 + X¹⁰⁰ + X²⁰⁰ + … + X⁹⁰⁰)(…)

Problem 23
Ann, Barbara, Carol, and Dorothy are members of the BobbySox Club. Every pair of members is together on one and only one committee. Each committee has exactly 3 members. What is the smallest possible total membership, and how many committees are there?

Problem 24
One of the largest known primes is 2³²¹⁷ – 1. Assume that it requires a human being a year to calculate each digit of this number. Who, if anyone, would have been capable of completing the job?

Problem 25
In the game of "Stogey", two players alternately place cigars on a rectangular table with the restriction that each new cigar must not touch any of the previously placed cigars. Can the 1st player assure himself of victory if we define the loser as the first player who finds himself without sufficient room to place a cigar?

Problem 26
Smith and Jones, both 50% marksmen, decide to fight a duel in which they exchange alternate shots until one is hit. What are the odds in favor of the man who shoots first?

Problem 27
Can 19₁₉ be represented as the sum of a cube and a fourth power?

Problem 28
How many primes are in the following infinite series where the digits are arranged in declining order? 9; 98; 987; 9876; …; 987654321; 9876543219; 98765432198; … etc.

Problem 29
What is the largest number which can be obtained as the product of positive integers which add up to 100?

Problem 30
The first expedition to Mars found only the ruins of a civilization. The explorers were able to translate a Martian equation as follows: 5x² – 50x + 125 = 0 x = {5, 8}. This was strange mathematics. The value x = 5 seemed legitimate enough but x = 8 required some explanation. If the Martian number system developed in a manner similar to ours, how many fingers would you say the Martians had?

Problem 31
A canoe is floating in a swimming pool. Which will raise the level of the water in the pool higher, dropping a penny into the pool or into the canoe? Or does it make any difference?

Problem 32
A rectangular picture, each of whose dimensions is an integral number of inches, has an ordinary rectangular frame 1 inch wide. Find the dimensions of the picture if the area of the picture and the area of the frame are equal.

Problem 33
A rook and a bishop are placed at random on different squares of a chessboard. What is the probability that one piece threatens the other?

Problem 34
Find the simplest solution in integers for the equation 1/X² + 1/Y² = 1/Z²

Problem 35
Five suspects where rounded up in connection with the famous "Red Robin Murder". Their statements were as follows: A: "C and D are lying." B: "A and E are lying." C: "B and D are lying." D: "C and E are lying." E: "A and B are lying." Who is lying?

Problem 36
Using the French Tricolor as a model, how many flags are possible with five available colors if two adjacent rows must not be colored the same?

Problem 37
In any gathering of six people prove that either three are mutually acquainted or three are mutually unacquainted.

Problem 38
Lottie and Lucy Hill are both 90 years old. Mary Jones, on the other hand, is half again as old as she was when she was half again as old as she was when she lacked 5 years of being half as old as she is now. How old is Mary?

Problem 39
What is the rightmost digit of 7⁷⁷?

Problem 40
Prove that the product of 4 consecutive positive integers cannot be a perfect square.

Problem 41
The game of reverse tic-tac-toe (known to some as toe-tac-tic) has the same rules as the standard game with one exception. The first player with three markers in a row loses. Can the player with the first move avoid being beaten?

Problem 42
A famous mountain climber was traveling through the Trondheim timber country one day. Quite by accident he dropped his trusty Alpenstock, an unusually straight stick, near the buzz saws where, in two shakes of a yak's tail, it was neatly cut into three pieces. What is the probability that these three pieces can be placed together to form a triangle?

Problem 43
A noted mathematician was shopping at a hardware store and asked the price of certain articles. The salesman replied. "One would cost 10 cents, eight would cost 10 cents, seventeen would cost 20 cents, one hundred and four would cost 30 cents, seven hundred and fifty six would also cost 30 cents, and one thousand and seventy two would cost 40 cents." What was the mathematician buying?

Problem 44
An elimination golf tournament is held for 53 players. How many golf matches will transpire?

Problem 45
Supply the missing number in the following sequence: 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, ?, 100, 121, 10,000.

Problem 46
Between Kroflite and Beeline are five other towns. The seven towns are an integral number of miles from each other along a straight road. The towns are spaced that if one knows the number of miles a person has traveled between any two towns he can determine the particular towns uniquely. What is the minimum distance between Kroflite and Beeline to make this possible?

Problem 47
A famous physicist who is always in a hurry, walks up an up-going escalator at the rate of one step per second. Twenty steps bring him to the top. Next day he goes up at two steps per second, reaching the top in 32 steps. How many steps are there in the escalator?

Problem 48
Two similar triangles with integral sides have two of their sides the same. The third sides differ by 387. What are the lengths of the sides?

Problem 49
A modernistic chess set has pieces in various geometrical shapes. In particular, both the KING and the KNIGHT are squares of integers. What numbers could these represent if each letter is replaced by a different digit?

Problem 50
In Byzatine basketball there are 35 scores which are impossible for a team to total, one of them being 58. Naturally a free throw is worth fewer points than a field goal. What is the point total of each?

Problem 51
What is larger, the tenth root of ten or the cube root of two?

Problem 52
The planet Octerra is divided into eight countries, each occupying an octant, thus each country borders three others. In how many ways can a traveler visit each of the other countries once and only once, returning to his home country only at the end of his trip?

Problem 53
A mathematician student was working on an assignment but, being a bit absent-minded, he forgot whether he was to add or to multiply the three different integers on his paper. He decided to do it both ways and, much to his surprise, the answer was the same. What where the three different integers.

Problem 54
Fourbisher and LaRouche started working for different firms at the same salary. Last year Fourbisher had a raise of 10%, and LaRouche had a drop in pay of 10%. This year Fourbisher had the 10% drop and LaRouche the 10% raise. Who is making more now?

Problem 55
A man has red, gray and black flagstones for making a walk. He wants no two consecutive stones to be the same color, no consecutive pair of stones to have the same two colors in the same order, no repetition of three consecutive colors, etc. He starts out laying first a red stone, then a gray, and continues until he finishes laying the seventh stone. He then finds himself stymied and unable to use any stone for the eighth without repetition of some color pattern. What were the colors of the first seven stones?

Problem 56
What letter follows OTTFFSSE_?

Problem 57
The sum and difference of two squares may be primes: 4 – 1 = 3 and 4 + 1 = 5; 9 – 4 = 5 and 9 + 4 = 13, etc. Can the sum and difference of two primes be squares? If so, for how many different primes is this possible?

Problem 58
In a certain community there are 1000 married couples. Two-thirds of the husbands who are taller than their wives are also heavier and three-quarters of the husbands who are heavier than their wives are also taller. If there are 120 wives who are taller and heavier than their husbands, how many husbands are taller and heavier than their wives?

Problem 59
The area and volume of a certain sphere are both 4-digit numbers times π. What is the radius of the sphere?

Problem 60
The numbers one through seven are drawn from a hat without replacement. What is the probability that all the odd numbers will be chosen first?

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