A mixed bag of questions that I’ve used for various vaguely similar purposes in the past:
(1) A suitably dressed-up version of the result that every graph must have an even number of odd vertices.
(2) There is a heap of $1001$ stones on the table. Repeat the following operation: choose some heap containing more than two stones, throw one stone away from that heap, and divide the remaining stones into two (not necessarily equal) heaps. Is it possible to end up with only heaps of size three?
(3) Let $n$ be an integer greater than $1$. The integers $1, 2, 3, \dots, n^2$ are placed on the squares of an $n\times n$ chessboard, one integer per square. Show that no matter how this is done, there must be two adjacent squares whose numbers differ by more than $n$. (The squares may be adjacent
horizontally, vertically, or diagonally.)
(4) Brynjulf and Kjellaug have three pieces of paper. At any time Brynjulf is allowed to pick up one piece of paper and tear it into three smaller pieces, and at any time Kjellaug is allowed to pick up one piece of paper and tear it into five smaller pieces; each of these operations is called a play. They are not allowed to play simultaneously, but they are also not required to take turns:
Brynjulf might make three plays in a row, then Kjellaug might make two, Brynjulf one, Kjellaug $17$, and so on. Can they manage to finish a play with exactly $100$ pieces of paper?
(5) Six bowls are arranged in a circle on the table, with one marble in each bowl; this is the starting position of a solitaire game. The rules are simple: at each turn you must pick up one marble in each hand and move it to one of the two bowls immediately adjacent to the one from which you got it. You are not required to draw the marbles from different bowls: if some bowl contains
more than one marble, you may take both from that bowl. If you do take two marbles from the same bowl, you need not move them to the same bowl: you may put one into each of the two neighboring bowls. The object of the game is to get all six marbles into a single bowl. Either explain how this can be done, or prove that it cannot be done.
(6) Variations on a theme: (a) Five darts are thrown at a square measuring $14$ inches on a side. Prove that two of them must be no more than $10$ inches apart. (b) Five darts are thrown at an equilateral triangle measuring $14$ inches on a side. Prove that two of them must be no more than $7$ inches apart. (c) Four darts are thrown at an equilateral triangle measuring $14$ inches on a side. Prove that two of them must be no more than $9$ inches apart. Can this maximum distance be improved to $8$ inches? (d) Nineteen darts are thrown at a regular hexagon measuring $12$ inches on a side. Prove that two of them must be no more than $7$ inches apart.
(7) Several people sit around a lunch table. As it happens, each person’s age is the average of the ages of the two people immediately to his or her right and left. Jessica says that she’s 26; Ian says that he’s only 24. How do I know that one of them is lying?