1. Restless frogs Why they can’t form a bigger square…

2. 1. Suppose the frogs are positioned in a unit square

3. 2. They can move only to unit grid points

7. I think it’s easy to see, can prove by induction…

8. 3. Now, suppose they succeeded to form a big square.

9. 4. Then they can reverse the steps and return to the original unit square formation.

10. 5. But… It’s impossible! Now they can move only along the coarse grid, so they never come back to the smaller square.

11. The same argument works for non-aligned squares…

13. What do you say? Is it convincing??

14. Here ’ s a riddle my supervisor heard from someone (more of an engineering nature): You are given two ropes and many matches. Each rope burns for exactly one hour. How would you measure 45 minutes with these tools only? (The two ropes might behave differently, and you cannot assume that the length of a rope is proportional to its burning time).