The Hidden Element

The name of one chemical element appears as an unbroken string in the names of four other elements. What is the element, and what are the four?

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A problem from the Leningrad Mathematical Olympiad: You have 32 stones, each of a different weight. How can you find the two heaviest in 35 weighings with an equal-arm balance?

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Express 1,000,000 as the product of two numbers, neither of which contains any zeroes.

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Match Point

A problem from the Leningrad Mathematical Olympiad: A and B take turns removing matches from a pile. The pile starts with 500 matches, A goes first, and the player who takes the last match wins. The catch is that the quantity that each player withdraws on a given turn must be a power of 2. Does either player have a winning strategy?

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From John Winter Jones’ Riddles, Charades, and Conundrums, 1822:

What is that which a coach always goes with, cannot go without, and yet is of no use to the coach?



From Routledge’s Every Boy’s Annual, 1864:

Why is O the noisiest of the vowels?

Because all the rest are inaudible (in audible).


From Mark Bryant’s Riddles: Ancient & Modern, 1983:

What is big at the bottom, little at the top, and has ears?

A mountain (it has mountaineers!).


piecework chess problems

Two ridiculous problems from Arthur Ford Mackenzie’s 1887 book Chess: Its Poetry and Its Prose:

Left: “White to mate without making a move.”

Right: “White to mate in 1/4 of a move.”

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The Savage Breast

A logic problem from Lewis Carroll: What conclusion follows from these premises?

  1. Nobody who really appreciates Beethoven fails to keep silent while the Moonlight Sonata is being played.
  2. Guinea pigs are hopelessly ignorant of music.
  3. No one who is hopelessly ignorant of music ever keeps silent while the Moonlight Sonata is being played.
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Seating Trouble

The Fall 1978 issue of Pi Mu Epsilon Journal included this problem, submitted by Pier Square. Four men are playing bridge. Their names are Banker, Waiter, Baker, and Farmer, and, as it happens, each man’s name is another man’s job. Mr. Baker’s partner is the baker, Mr. Banker’s partner is the farmer, and the waiter sits at Mr. Farmer’s right. Who is sitting at the banker’s left?

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Seeing Red

University of Waterloo mathematician Ross Honsberger chose this problem for his 2004 collection Mathematical Delights; it’s a generalization of a problem that Robert Gebhardt had offered in the Fall 1999 issue of Pi Mu Epsilon Journal. Paint the outside of an n × n × n cube red, then chop it into n3 unit cubes. Put the unit cubes in a box, mix them up thoroughly, withdraw one at random, and throw it across a table. What’s the probability that it comes to rest with a red face on top?

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