Mixed Emotions

A brainteaser by S. Ageyev, from the November-December 1991 issue of Quantum:

ageyev problem

Suppose that we change the signs of 50 of these numbers such that exactly half the numbers in each row and each column get a minus sign. Prove that the sum of all the numbers in the resulting table is zero.

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Riddle

When Louis Philippe was deposed, why did he lose less than any of his subjects?

Because, while he lost only a crown, they lost a sovereign.

— Edith Bertha Ordway, The Handbook of Conundrums, 1915

Birds of a Feather

A problem from the February 2006 issue of Crux Mathematicorum:

Prove that if 10a + b is a multiple of 7 then a – 2b must be a multiple of 7 as well.

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Readouts

sallows readouts puzzle

A puzzle by Lee Sallows. In this readout from a computer-driven electronic display, the digits in the fifth row have been obscured. What are they?

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Staffing

A problem from Crux Mathematicorum, April 2006:

A group of people must be formed into committees. Show that the number of possible committees that can be formed with an odd number of members is the same as the number that can be formed with an even number of members. (Assume that a committee with no members and one that includes everyone are both allowed.)

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Can Do

https://archive.org/details/samloydcyclopedi00libg/page/n25

A puzzle by Sam Loyd:

John the milkman has two 10-gallon cans full of milk. Two customers have a 5- and a 4-quart measure and want 2 quarts put into each measure. How can he accomplish this?

“It is a juggling trick pure and simple, devoid of trick or device, but it calls for much cleverness to get two exact quarts of milk into those measures employing no receptacles of any kind except the two measures and the two full cans.”

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An Observant Anthropologist

A puzzle from the 1998 Moscow Mathematical Olympiad, via Peter Winkler’s excellent Mathematical Puzzles, 2021:

An anthropologist is surrounded by a circle of natives. Each native either always lies or always tells the truth. The anthropologist asks each native whether the native to his right is a liar or a truth teller. From their answers, she’s able to deduce the fraction of the circle who are liars. What is the fraction?

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Two Weighings

A problem from the Leningrad Mathematical Olympiad: You have a set of 101 coins, and you know that it contains one counterfeit coin X. The 100 genuine coins all have the same weight, which is different from that of X. Using only two weighings in an equal-arm balance, how can you determine whether X is heavier or lighter than the genuine coins?

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