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.

July 23, 2012February 5, 2013 | Puzzles

Tall and Wide

tall and wide 1

Which of these triangles has the greater area?

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Each can be decomposed into two 3-4-5 triangles … so they’re equal.

July 21, 2012February 5, 2013 | Puzzles

Black and White

van beek chess problem

By H. Van Beek, Chemnitzer Wochenschach, 1926 (special prize). White to mate in two moves.

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1. Nc4 and White will mate by 2. d5, 2. Qg8, or 2. Qc8.

July 20, 2012February 5, 2013 | Puzzles

Outpourings

gedanken tank puzzle 1

A tank of water has two holes of equal area, one at top and one at bottom. The top one leads to a downspout, so that both holes discharge their water at the same level. Ignoring friction, which hole produces the faster flow of water?

You actually don’t need to know the physics in order to solve this — it yields to an insight.

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The water flows from both openings at the same rate. If it didn’t, we could connect the two to create a perpetual-motion machine. (From Lewis Epstein’s Thinking Physics Is Gedanken Physics, 1989.)

July 19, 2012February 5, 2013 | Puzzles · Science & Math