A nice story problem from Germany
(Choo Ray here.) I thought it may be a good idea to write about some of the more interesting problems that I have used in my recent sessions for students in the Singapore National Team.
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Painting by Numbers
(This is Lucas.) Hi guys. It's been a long couple weeks (for those who are still active students, at least), let's wind down with some combi. In this post, I'll try to (at least partially) de-mystify colouring/invariants. As always, I'd encourage those of you who are still students to try the questions yourself to get a feel for them before reading the explanations that follow. Question 1 Each cell of a 726 by 726 grid is initially filled with a "$+1$". In a move, we may flip the signs of a 3 by 3 grid of our choice, less two opposite corners (so we flip 7 cells in each move). Can we reach a state where every cell contains a "$-1$"? Effectively, we affect 7 cells in this formation (or its reflection). Initial Thoughts The setup is practically begging us to apply some colouring. However, the usual suspects don't yield any results, and thinking broadly it's hard to imagine a coloring applying nicely to a set of 7 cells. Maybe we can pla...