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«Contents Ü Foreword Elwyn Berlekamp and Tom Rodgers ½ I Personal Magic ¿ Martin Gardner: A “Documentary” Dana Richards ½¿ Ambrose, Gardner, ...»

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I thank the following people for providing problems and ideas that have contributed to the richness of the problems involved: Leon Bankoff, Brian Barwell, Nick Baxter, Laurie Brokenshire, James Dalgety, Clayton Dodge, Martin Gardner, Dieter Gebhardt, Allan Gottlieb, Yoshiyuki Kotani, Harry Nelson, Karl Scherer, David Singmaster, Naoaki Takashima, Dario Uri, Bob Wainwright, and, especially, Nob Yoshigahara. Source information begins on page 82.

Easy Problems

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E4. It is approximately 2244.5 nautical miles from Los Angeles to Honolulu. A boat starts from being at rest in Los Angeles Harbor and proceeds at 1 knot per hour to Honolulu. How long does it take?

E5. You have containers that hold 15 pints, 10 pints, and 6 pints. The 15pint container starts out full, and the other two start out empty: (15, 0, 0). Through transferring liquid among the containers, measure exactly two pints for yourself to drink and end up with 8 pints in the 10-pint container and 5 pints in the 6-pint container. Find the most efficient solution.

E6. You are at a lake and have two empty containers capable of holding exactly ´ ¿ ½ ½ µ and ´ ¾ ½ ¾ ½ µ liters of liquid. How many transfers of liquid will it take you to get a volume of liquid in one container that is within one percent of exactly one liter?

E7. Calculate the expansion of this 26 term expression:

´Ü   µ´Ü   µ´Ü   µ´Ü   µ ´Ü   ݵ´Ü   Þµ

. 8. Find a nine-digit number made up of 1, 2, 3, 4, 5, 6, 7, 8, 9 in some perE mutation such that when digits are removed one at a time from the right the number remaining is divisible in turn by 8, 7, 6, 5, 4, 3, 2 and 1.

E9. My regular racquetball opponent has a license plate whose three-digit part has the following property. Divide the number by 3, reverse the digits of the result, subtract 1 and you produce the original number. What is the number and what is the next greater number (possibly with more than three digits) having this property?


E10. Find nine single-digit numbers other than (1, 2, 3,..., and 9) with a sum of 45 and a product of 9 362,880.

E11. A knight is placed on an infinite checkerboard. If it cannot move to a square previously visited, how can you make it unable to move in as few moves as possible?

E12. Place the numbers from 1 to 15 in the 3 ¢ 5 array so that each column has the same sum and each row has the same sum.

E13. The Bridge. Four men must cross a bridge. They all start on the same side and have 17 minutes to get across. It is night, and they need their one flashlight to guide them on any crossing. A maximum of two people can cross at one time. Each man walks at a different speed: A takes 1 minute to cross; B takes 2 minutes; C takes 5 minutes, and D takes 10 minutes. A pair must walk together at the rate of the slower man’s pace. Can all four men cross the bridge? If so, how?

Try these other problems.

(a) There are six men with crossing times of 1, 3, 4, 6, 8, and 9 minutes and they must cross in 31 minutes.

(b) There are seven men with crossing times of 1, 2, 6, 7, 8, 9, and 10 minutes, and the bridge will hold up to three men at a time, and they must cross in 25 minutes.

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E15. Potato Curves. You are allowed to draw a closed path on the surface of each of the potatoes shown below. Can you draw the two paths so that they are identical to each other in three-dimensional space?

E16. Suppose a clock’s second hand is exactly on a second mark and exactly 18 second marks ahead of the hour hand. What time is it?

E17. Shoelace Clock. You are given some matches, a shoelace, and a pair of scissors. The lace burns irregularly like a fuse and takes 60 minutes to burn from end to end. It has a symmetry property in that the burn rate a distance Ü from the left end is the same as the burn rate the same distance Ü from the right end. What is the minimum time interval you can measure?

Medium Problems

M1. You have a 37-pint container full of a refreshing drink. N thirsty customers arrive, one having an 11-pint container and another having a 2Npint container. How will you most efficiently measure out 1 pint of drink for each customer to drink in turn and end up with N pints in the 11-pint container and 37-2N pints in the 37-pint container if

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M2. Three points have been chosen randomly from the vertices of a cube.

What is the probability that they form (a) an acute triangle; (b) a right triangle?

M3. You were playing bridge as South and held 432 in spades, hearts, and diamonds. In clubs you held 5432. Despite your lack of power you took 6 tricks, making a 4-club contract. Produce the other hands and a line of play that allows this to occur.

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M8. The river shown below is 10 feet wide and has a jog in it. You wish to cross from the south to the north side and have only two thin planks of length L and width 1 ft to help you get across. What is the least value for L that allows a successful plan for crossing the river?

M9. A regular pentagon is drawn on ordinary graph paper. Show that no more than two of its vertices lie on grid points.

M10. 26 packages labeled A to Z are known to each weigh whole numbers of pounds in the range 1 to 26.

(a) Determine the weight of each package with a two-pan balance and four weights of your own design.

(b) Now do it with three weights.

M11. Music on the planet Alpha Lyra IV consists of only the notes A and B.

Also, it never includes three repetitions of any sequence nor does the repetition BB ever occur. What is the longest Lyran musical composition?

M12. Many crypto doorknob locks use doors with five buttons numbered from 1 to 5. Legal combinations allow the buttons to be pushed in specific order either singly or in pairs without pushing any button more than once.

Thus [(12), (34)] = [(21), (34)]; [(1), (3)]; and [(2), (13), (4)] are legal combinations while [(1), (14)]; [(134)]; and [(13), (14)] are not.

(a) How many legal combinations are there?

(b) If a sixth button were added, how many legal combinations would there be?

M13. Find the smallest prime number that contains each digit from 1 to 9 at least once.

58 R. I. HESS M14. Dissect a square into similar rectangles with sides in the ratio of 2 to 1 such that no two rectangles are the same size. A solution with nine rectangles is known.

M15. Divide an equilateral triangle into three contiguous regions of identical shape if (a) All three regions are the same size;

(b) all three regions are of different size;

(c) two of the regions are the same size and the third region is a different size.

M16. Dissect a square into similar right triangles with legs in the ratio of 2 to 1 such that no two triangles are the same size. A solution with eight triangles is known.

M17. You and two other people have numbers written on your foreheads.

You are told that the three numbers are primes and that they form the sides of a triangle with prime perimeter. You see 5 and 7 on the other two people, both of whom state that they cannot deduce the number on their own foreheads. What number is written on your forehead?

M18. A snail starts crawling from one end along a uniformly stretched elastic band. It crawls at a rate of 1 foot per minute. The band is initially 100 feet long and is instantaneously and uniformly stretched an additional 100 feet at the end of each minute. The snail maintains his grip on the band during the instant of each stretch. At what points in time is the snail (a) closest to the far end of the band, and (b) farthest from the far end of the band?

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M20. Humpty Dumpty. “You don’t like arithmetic, child? I don’t very much,” said Humpty Dumpty.

“But I thought you were good at sums,” said Alice.

“So I am,” said Humpty Dumpty. “Good at sums, oh certainly. But what has that got to do with liking them? When I qualified as a Good Egg — many, many years ago, that was — I got a better mark in arithmetic than any of the others who qualified. Not that that’s saying a lot. None of us did as well in arithmetic as in any other subject.” “How many subjects were there?” said Alice, interested.

“Ah!” said Humpty Dumpty, “I must think. The number of subjects was one third of the number of marks obtainable in any one subject. And I ought to mention that in no two subjects did I get the same mark, and that is also true of the other Good Eggs who qualified.” “But you haven’t told me,” began Alice.

“I know I haven’t told you how many marks in all one had to obtain to qualify. Well, I’ll tell you now. It was a number equal to four times the maximum obtainable in one subject. And we all just managed to qualify.” “But how many,” said Alice.

“I’m coming to that,” said Humpty Dumpty. “How many of us were there? Well, when I tell you that no two of us obtained the same assortment of marks — a thing which was only just possible — you’ll be well on the way to the answer. But to make it as easy as I can for you, I’ll put it another way. The number of other Good Eggs who qualified when I did, multiplied by the number of subjects (I’ve told you about that already), gives a product equal to half the number of marks obtained by each Good Egg. And now you can find out all you want to know.” He composed himself for a nap. Alice was almost in tears. “I can’t,” she said, “do any of it. Isn’t it differential equations, or something I’ve never learned?” Humpty Dumpty opened one eye. “Don’t be a fool, child,” he said crossly. “Anyone ought to be able to do it who is able to count on five fingers.” What was Humpty Dumpty’s mark in Arithmetic?

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H2. “To reward you for killing the dragon,” the Queen said to Sir George, “I grant you the land you can walk around in a day.” She pointed to a pile of wooden stakes. “Take some of these stakes with you,” she continued.

“Pound them into the ground along your way, and be back at your starting point in 24 hours. All the land in the convex hull of your stakes will be yours.” (The Queen had read a little mathematics.) Assume that it takes Sir George 1 minute to pound a stake and that he walks at a constant speed between stakes. How many stakes should he take with him to get as much land as possible?

H3. An irrational punch centered on point P in the plane removes all points from the plane that are an irrational distance from P. What is the least number of irrational punches needed to eliminate all points of the plane?

H4. Imagine a rubber band stretched around the world and over a building as shown below. Given that the width of the building is 125 ft and the rubber band must stretch an extra 10 cm to accommodate the building, how tall is the building? (Use 20,902,851 ft for the radius of the earth.) H5. A billiard ball with a small black dot, P, on the exact top is resting on the horizontal plane. It rolls without slipping or twisting so that its contact point with the plane follows a horizontal circle of radius equal to that of the ball. Where is the black dot when the ball returns to its initial resting place?


–  –  –

H9. As shown below it is easy to place 2n unit diameter circles in a 2 ¢ n rectangle. What is the smallest value of n for which you can fit 2n + 1 such circles into a 2 ¢ n rectangle?

H10. You are given two pyramids SABCD and TABCD. The altitudes of their eight triangular faces, taken from vertices S and T, are all equal to 1.

Prove or disprove that line ST is perpendicular to plane ABCD.

–  –  –

H12. My uncle’s ritual for dressing each morning except Sunday includes a trip to the sock drawer, where he (1) picks out three socks at random, then (2) wears any matching pair and returns the odd sock to the drawer or (3) returns the three socks to the drawer if he has no matching pair and repeats steps (1) and (3) until he completes step (2). The drawer starts with 16 socks each Monday morning (eight blue, six black, two brown) and ends up with four socks each Saturday evening.

(a) On which day of the week does he average the longest time to dress?

(b) On which day of the week is he least likely to get a pair from the first three socks chosen?

H13. The hostess at her 20th wedding anniversary party tells you that the youngest of her three children likes her to pose this problem, and proceeds to explain: “I normally ask guests to determine the ages of my three children, given the sum and product of their ages. Since Smith missed the problem tonight and Jones missed it at the party two years ago, I’ll let you off the hook.” Your response is “No need to tell me more, their ages are ” H14. Minimum Cutting Length.What is the minimum cut-length needed to divide (a) a unit-sided equilateral triangle into four parts of equal area?

(b) a unit square, into five parts of equal area?

(c) an equilateral triangle into five parts of equal area?

–  –  –

H16. (a) The figure shows two Pythagorean triangles with a common side where three of the five side-lengths are prime numbers. Find other such examples.

–  –  –

H17. The inhabitants of Lyra III recognize special years when their age is of Ô¾Õ, where Ô and Õ are different prime numbers. The first few the form such special years are 12, 18 and 20. On Lyra III one is a student until reaching a special year immediately following a special year; one then becomes a master until reaching a year that is the third in a row of consecutive special years; finally one becomes a sage until death, which occurs in a special year that is the fourth in a row of consecutive special years.

(a) When does one become a master?

(b) When does one become a sage?

(c) How long do the Lyrans live?

(d) Do five or six special years ever occur consecutively?

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