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Found: 68

Does there exist a polygon intersecting each of its own sides only once (each side is intersected only once by a different side) and has all together (a) 6 sides; (b) 7 sides.

Is it possible to make a hole in a wooden cube in such a way that one can drag another cube of the same size through that hole?

Is it true that for any point inside any convex quadrilateral the sum of the distances from the point to the vertices of the quadrilateral is less than the perimeter?

The March Hare bought seven drums of different sizes and seven drum sticks of different sizes for his seven little leverets. If a leveret sees that their drum and their drum sticks are bigger than a sibling’s, they start drumming as loud as they can. What is the largest number of leverets that may be drumming together?

Mary Ann and Alice went to buy some cupcakes. There are at least five different types of cupcakes for sale (all different types are priced differently). Mary Ann says, that whatever two cupcakes Alice buys, Mary Ann can always buy another two cupcakes spending the same amount of money as Alice. What should be the smallest number of cupcakes available for sale at the shop if Mary Ann is not lying?

A natural number \(n\) can be exchanged to number \(ab\), if \(a+b=n\) and \(a\) and \(b\) are natural numbers. Is it possible to receive 2017 from 22 after such manipulations?

Alice marked several points on a line. Then she put more points – one point between each two adjacent points. Show that the total number of points on the line is always odd.

The Dormouse brought a \(4\times 10\) chocolate bar to share at the tea party. She needs to break the bar by the lines into single pieces (without any lines on it). In one turn she can cut one piece into two along the lines. What is the least number of cuts she needs to make to break the bar into single pieces?

Tweedledum and Tweedledee travel from their home to the castle of the White Queen. Having only one bicycle among them, they take turns in riding the bike. While one of them is cycling, the other one walks (walking means walking and not running). Nevertheless, they manage to arrive to the castle nearly twice as fast in comparison to if they both were walking all the way to the castle. How did they manage it?

There are \(n\) inhabitants (\(n>3\)) in the Wonderland. Each habitant has a secret, which is known to him/her only. In a telephone conversation two inhabitants tell each other all the secrets they know. Show that after \((2n-4)\) conversations all the secrets may be spread among all the inhabitants.