Can a cube be split into finely many smaller cubes, all with different sizes?
SOLUTION
It is impossible. Assume the opposite and consider the smallest cube on the bottom, C1. Since all cubes surrounding it are larger, they must “wall in” its top face. We repeat the same argument for the smallest cube which lies on the top face of C1, call it C2. If all surrounding cubes are larger, they must wall in its top face. Thus, we can create an infinite sequence of cubes with decreasing sizes lying on top of each other: C1, C2, C3, etc. Since the initial cube is split into a finite number of smaller cubes, we get a contradiction.
A Boeing 747 is sitting on a conveyor belt, as wide and long as a runway. The conveyor belt is designed to exactly match the speed of the wheels, moving in the opposite direction. Will the plane take off?
A teacher in English had asked James and John to describe a man who had suffered from a cold in the past. James while John had had had had had had had had had had had a better effect on the teacher.
Add punctuation to the sentence in bold, so that it makes sense.
SOLUTION
A teacher in English had asked James and John to describe a man who had suffered from a cold in the past. James, while John had had “had”, had had “had had”; “had had” had had a better effect on the teacher.
The meaning is that John had used the phrase “had a cold”, whereas James had used the phrase “had had a cold”. The latter, being more grammatical, had resulted in a better impression on the teacher.
Cut out the black shape and then fold it perfectly into a cube, without overlapping.
SOLUTION
If the size of each of the small edges of the shape is equal to 1cm, then its area is equal to 30cm². Thus, the cube that is formed by the shape must have an edge of length √5cm. If we set the center of the black shape to be the center of one of the cube’s faces, then the four closest vertices must lie on the boundary of the shape, √2.5cm away. This determines uniquely the folding of the shape which is shown on the simulation below.
Three friends, A, B and C, want to find out what their average salary is without disclosing their own salaries to the others. How can they do it using only verbal communication?
SOLUTION
A tells B some number, then B adds his salary to it and tells the result to C, then C adds his salary and tells the result to A. Now A subtracts the number he told B in the beginning, adds his own salary and divides by 3. Repeat the same procedure with B and C starting first.
Help Mrs. Vampire get to her sweet home safely, avoiding all the light coming from outside. Of course, she can close the doors if she gets close enough to them.
Six identical glasses are placed in a row on the table – first three filled with water, and then three empty ones. Can you move just one glass, so that empty and full glasses alternate?
SOLUTION
Take the second full glass, pour all the water into the second empty glass, and then put it back in its place.