Teen kids are in the forest picking mushrooms. No one came home empty-handed, together they collected 54 mushrooms. Show that at least two children picked the same amount of mushrooms.
Paul has a rectangular patio which is covered with concrete slabs. Some of the plates in the form of 2x2, other form of 1x4.One of the plates has been broken, but as a possible replacement Paul only have a single plate of the other form. Can Paul replace the broken plate with the one he has, or possibly rearrange the tiles?
If two of them did not pick the same amount of mushrooms, then they must have at least collected 1,2,3...,10 mushrooms, with one mushroom coresponding to one child. But that gives a total of at least 55 mushrooms, a contradiction.
For the second one, it cannot be done. This is because, suppose that the 2×2 tile has been broken. Then replacing it with one of the 1×4 tiles changes the parity of the dimension across which the 1 width tile is placed. In that case, we cannot compensate for this by rearrangement. Likewise vice versa.