Desperate Measuring

desperate measuring

You’re alone on a desert island and want to lay out a course for some snail races. Unfortunately, you have only an 8.5 x 11 inch sheet of paper. How can you use it to measure exactly 3 inches?

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Here are two ways: “Dog-ear” the page, folding one corner across to the opposite side. The area below the dog-eared portion measures 8.5 x 2.5. Reopen the paper and fold this 8.5 x 2.5 portion up into the page. The area that remains uncovered measures 8.5 x 6; it can be folded in half to produce a height of 3 inches. Fold the paper in half to make a doubled sheet measuring 5.5 x 8.5. Now “dog-ear” this doubled page, folding one corner across to the opposite side. The area below the dog-eared portion measures 5.5 x 3.

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August 6, 2012February 6, 2013 | Puzzles

The Last Cent

You and a friend are playing a game. Between you is a pile of 15 pennies. You’ll take turns removing pennies from the pile — each of you, on his turn, can choose to remove 1, 2, or 3 pennies. The loser is the one who removes the last penny.

You go first. How should you play?

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Consider what happens if you leave your opponent 5 pennies. At that point, no matter how many he removes, you can play so as to leave the final penny to him. Now, if leaving him 5 pennies is a worthy target, then by the same principle so is leaving him 9 pennies: No matter how he plays, he must now give you the opportunity to leave him with 5. And going one step further, aiming for 13 pennies will ensure that you can reach 9, then 5, and victory. So, on your first turn, take 2 pennies.

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August 5, 2012February 6, 2013 | Puzzles

Fish Story

I release a fish at the edge of a circular pool. It swims 80 feet in a straight line and bumps into the wall. It turns 90 degrees, swims another 60 feet, and hits the wall again. How wide is the pool?

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It sounds as if there’s not enough information to solve this, but there is if you remember that when a right triangle is inscribed in a circle, its hypotenuse is the diameter. The pool is 100 feet wide.

August 4, 2012February 5, 2013 | Puzzles

Black and White

fleischmann chess problem

By Bertha Fleischmann. White to mate in two moves.

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1. Qa8 and the queen will take the pawn, giving mate.

August 3, 2012February 5, 2013 | Puzzles

Changing Colors

Fifty-five chameleons live on a tropical island. Thirteen are green, 19 are brown, and 23 are gray. Whenever two chameleons of different colors meet, both change to the third color. Is it possible that all 55 chameleons might eventually be the same color?

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Yes. Let x, y, and z be the number of green, brown, and gray chameleons, and consider the remainders left when each of the quantities x-y, x-z, and y-z is divided by 3. These remainders won’t change as the chameleons vary their colors. This means that if two populations are to reach 0 simultaneously, they must differ now by a multiple of 3. In this case two do: The populations of green and brown chameleons differ by 6, so it’s possible for both to disappear without any leftovers: 13, 19, 23 15, 18, 22 17, 17, 21 … and then green and brown chameleons can disappear in pairs until all have turned gray. From Dick Hess, All-Star Mathlete Puzzles, 2009.

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July 31, 2012February 5, 2013 | Puzzles

A Bit More

What’s the longest month of the year in London?

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In the U.K., October is one hour longer than any other month because of the end of Daylight Saving Time. See Fugue State.

July 30, 2012February 5, 2013 | Puzzles

Wild Kingdom

Fourteen ladders stand in a row. At the foot of each ladder is a monkey; at the top is a banana. Festooning the ladders are an arbitrary number of ropes. Each rope connects a rung on one ladder to a rung on another, but no rung receives more than one rope.

At a signal all 14 monkeys begin climbing. If a monkey encounters a rope it climbs along it to the other end and then continues climbing upward. Show that every monkey gets a banana.

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Imagine the process in reverse. If Monkey 1 reaches Banana G and then descends again according to the same rules, it will return to the foot of Ladder 1; its path is uniquely determined. Further, because there’s no occasion for loops or dead ends, each monkey will eventually reach the top of some ladder. If every monkey reaches a goal, and no two monkeys reach the same goal, then every monkey gets a banana. The solution generalizes to any number of ladders.

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July 28, 2012February 5, 2013 | Puzzles

Black and White

By F.C. Collins. White to mate in two moves.

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1. After the sneaky 1. Qa8 Black can choose among six different mates.

July 27, 2012February 5, 2013 | Puzzles

Unwound

You have 13 reels of magnetic tape, one empty reel, and a machine that will wind tape from a full reel to an empty one, reversing its direction. You need all 13 tapes reversed on their original reels. Show how this can be done, or prove that it’s impossible.

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It’s impossible. Each of the 13 reels must go through an even number of transfers in order to finish with its original tape. (Likewise, the empty reel must go through an even number of transfers to remain empty.) But each of the 13 tapes must be transferred an odd number of times to end up reversed. So the total number of transfers must be both even and odd, an impossibility.