Forefathers

A problem from the 1996 mathematical olympiad of the Republic of Moldova:

Twenty children attend a rural elementary school. Every two children have a grandfather in common. Prove that some grandfather has not less than 14 grandchildren in this school.

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Red or Black

Six sheets are set out in a room. Each identifies a different date in the same month. Weekdays are printed in black and Sundays in red. Six people will enter the room, one by one. Before the first one enters, one sheet is turned face down. Candidate 1 is then asked if she can deduce the color of the inverted sheet by examining the other sheets. Her answer, yes or no, is written on the back of the inverted sheet, followed by her number, 1. When 1 departs, a second sheet is turned face down. Candidate 2 enters and is asked whether she can deduce the color of the second sheet by considering her predecessor’s answer and the four face-up sheets. Her answer is noted in its turn, and this process continues — when the sixth candidate enters, she sees six face-down sheets, the first five of which bear the answers of the first five candidates. If all five of these answers are no, can Candidate 6 answer yes?

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Black and White

vielväter problem

In 1949 R.J. Darvall presented this position with a simple question: Who wins?

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The Ranchers’ Split

https://pxhere.com/en/photo/626898

A problem from the Graham Dial, a publication of Graham Transmissions Inc., via L.A. Graham’s The Surprise Attack in Mathematical Problems (1968):

Little Euclid went with his father to visit a friend who ran a small ranch. The rancher told them that he and his partner used to herd cattle but had then sold the herd for as many dollars per head as there were head in the herd. With the proceeds they had bought as many sheep as they could at $10.00 per sheep, and with the money that remained they bought a dog. Later the rancher and his partner decided to split up evenly, so they added the dog to the herd and each of them took the same number of animals.

Euclid surprised him by saying, “I hope you paid him $2.00 to even things up.”

“Why, yes, I did,” the rancher said, “but how did you know? I haven’t told you any of the numbers or prices.” Can you prove that Little Euclid had to be correct?

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Next Up

What’s the next number in this sequence?

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, …

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Custom Baking

From a Russian puzzle collection:

Is it possible to bake a cake that can be divided into four parts by a single straight cut?

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Black and White

orbán chess puzzle

Tibor Orbán offered this puzzle in Die Schwalbe in 1976. The position above can be reached in exactly 3 moves in several ways — for example:

1. e4 c6 2. Bb5 e6 3. Bxc6 dxc6
1. e4 e6 2. Bc4 c6 3. Bxe6 dxe6

How can it be reached in exactly 4 moves?

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Trip Planning

mouse puzzle

A mouse wants to eat his way through a 3 × 3 × 3 cube of cheese, starting in one of the corners and tunneling through all 27 1 × 1 × 1 sub-cubes, visiting each once. Can he arrange his route so that he finishes at the center of the cube? Assume that he always moves between orthogonally adjacent cubes, traveling through walls but not through edges or corners.

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