Buried Up to Neck

Section: Brain Teasers Category: Deduction
Difficulty:

Three friends, Adam, Bob, and Charlie are buried in the sand up to their necks, all facing West. Charlie can see both Adam and Bom, Bom can see only Adam, and Adam cannot see anyone. Black and white hats are placed on their heads. The three friends are told that there is at least one hat from each color, and then they are asked whether anyone can guess the color of their own hat.

After a few minutes, one of them answers. Who is that?

If Adam and Bob had hats with identical colors, then Charlie would immediately be able to deduce that his hat has the opposite color. Charlie doesn’t do that, so Adam and Bob are able to figure out that their hats have opposite colors. Since Bob is the one who can see the color of Adam’s hat, he is the one that answers the question.

unknown-author
Unknown Author February 13, 2024
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Fish Eat Fish

Section: Brain Teasers Category: Mathematics
Difficulty: Tags:

A hundred fish are swimming along a stream at different velocities. If one fish catches up to another fish, it eats it and continues swimming. What is the expected number of fish that will survive?

Notice that the N-th fish survives if and only if it is the fastest among the first N fish. The probability of this event is equal to 1/N. Since the expected number of fish that survive is equal to the sum of the survival probabilities for each of them, the answer is 1+1/2+1/3+…+1/N.

unknown-author
Unknown Author February 7, 2024
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Worm in an Apple

Section: Brain Teasers Category: Mathematics
Difficulty: Tags:

There is a perfectly spherical apple with a radius 50mm. A worm has entered the apple, made a tunnel of length 99mm through it and left. Prove that we can slice the apple in two pieces through the center, so that one of them is untouched by the worm.

Let the entering point is A, the leaving point is B and the center of the apple is C. Consider the plane P containing the points A, B and C and project the worm’s tunnel on it. Since 99 < 2×50, the convex hull of the tunnel’s projection will not contain the center C. Therefore we can find a line L through C, such that the tunnel’s projection is entirely in one of the semi-planes of P with respect to L. Now cut the apple with a slice orthogonal to P passing through the line L and you are done.

unknown-author
Unknown Author February 1, 2024
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Slicing Butter

Section: Brain Teasers Category: Insight
Difficulty:

If you want to split a cubic piece of butter into 27 smaller cubes, you can easily do it using just 6 slices (imagine the Rubik’s Cube). However, after every slice you make you can also rearrange the pieces – stack them in different ways on top of each other so that the number of cuts possibly gets reduced. What is the minimum number of slices you need in order to accomplish the task?

In order to separate the little cube in the center of the butter piece from the rest, you need 6 slices – that’s the number of sides it has. Therefore, you can’t accomplish the task with less than 6 cuts.

unknown-author
Unknown Author January 22, 2024
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