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A string is wound around a circular rod with circumference 10 cm and length 30 cm. If the string goes around the rod exactly 4 times, what is its length?
Imagine the circular rod is actually a paper roll and the string is embedded inside the paper. When we unroll it, we get a paper rectangle 30cm×40cm with the string embedded along the diagonal. Using the Pythagorean theorem, we find that the length of the string is 50cm.
Two friends, logicians – Ein and Stein – get imprisoned in two distant cells in a castle. Both cells have just one door, and a window with 8 bars in the first cell, and 12 bars in the second cell. The first day both logicians get the same letter from the prison master:
“The total number of bars in the two prison cells in this castle is either 18 or 20. Starting tomorrow, every morning I will go first to Ein and then to Stein, and will ask how many bars the other logician has. If one of you answers correctly, I will immediately let both of you leave the castle. If one of you answers incorrectly, I will execute both of you. Of course, you can always decide not to answer and just stay imprisoned.
I have sent a copy of this letter to you and your friend. There is no point in trying to communicate with him – your cells are far away from each other, and he won’t hear you.”
Will the logicians manage to escape the castle eventually? When will they do it?
Solution coming soon.
9 8 7 6 5 4 3 2 1 = 1 0 0
Add 3 pluses and 1 minus (not necessarily in that order) anywhere above in order to get a valid equality.
The answer is:
9 8-7 6+5 4+3+2 1 = 1 0 0
You are lost in the middle of a forest, and you know there is a straight road exactly 1 km away from you, but not in which direction. Can you find a path of distance less than 640 m which will guarantee you to find the road?
Imagine there is a circle with a radius of 100 m around you, and you are at its center O. Let the tangent to the circle directly ahead of you be t. Then, follow the path:
An ant is positioned at one of the vertices of a cube and wants to get to the opposite vertex. If the edges of the die have length 1, what is the shortest distance the ant needs to travel?
We unfold a cube to get a cross-shaped figure. Then, the problem is to find the shortest path between two points separated by a horizontal distance of 2 units and a vertical distance of 1 unit.
It is easy to see that the path in question is the one passing through the middle of the edge between the start and end points, and which has a distance of √5.
What should follow the sequence of diagrams shown below?
The diagrams represent consecutive monarchs of England:
Therefore, the next diagram should represent King Charles the 3-rd → Kc3.
There is a square drawn on a piece of paper and also a point marked with invisible ink. You are allowed to draw 3 lines on the paper and for each of them you will be told whether the point is on its left, on its right, or lies on the line. Your task is to find out whether the point is inside the square, outside the square, or on its boundary. How do you do it?
Draw one of the diagonals of the square. Then, draw the 2 lines containing the sides of the square that are on the same side as the invisible point.
White to play and mate in 2.
Note: Magnus Carlsen spent almost 2 minutes solving this puzzle.
The first move is 1. Qa8. If Black plays Qd4, then White responds with Re3#. Otherwise, White plays either Nc5# or Nd6#, so that Black cannot capture the Knight neither with the Rook or the Queen.
For this puzzle/game, you need to keep presenting various words to your friends and telling them whether they are “TOUCH” or they are “DON’T TOUCH”. Below, we have listed several words from each category.
TOUCH: banana, proof, mouse, keyboard, promo, woman
DON’T TOUCH: cherry, solution, cat, screen, discount, girl
Can you guess what determines whether a word is “TOUCH” or “DON’T TOUCH”?
Words that make your lips touch when pronounced belong to the “TOUCH” category, while the others belong to the “DON’T TOUCH” category. The sounds “P”, “B”, “M”, and “W” that cause this are called “bilabial”.
Five points, A, B, C, D, and E, lie on a line. The distances between them in ascending order are: 2, 5, 6, 8, 9, X, 15, 17, 20, and 22. What is X?
We assume that the points are ordered A to E from left to right. We have AE = 22 and either AD = 20, BE = 17, or AD = 17, BE = 20. Without loss of generality AD = 20, BE = 17, and therefore AB = 5, BD = 15, DE = 2. The distance of 6 is associated with either BC or CD, and therefore the points are arranged in one of these two ways:
If it is the latter, we get the sequence of distances: 2, 5, 6, 9, 11, 14, 15, 17, 20, 22, which does not fit the provided sequence.
If it is the former, we get the sequence of distances: 2, 5, 6, 8, 9, 14, 15, 17, 20, 22, and therefore X = 14.
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