Mutilated chessboard problem(en.wikipedia.org)
en.wikipedia.org
Mutilated chessboard problem
https://en.wikipedia.org/wiki/Mutilated_chessboard_problem
4 comments
Yes - there is a simple, clear, elegant way to see that removing one square of each colour leaves the board coverable. It's even given in the article.
I got the correct answer immediately because I've seen a similar problem before in HN, although with a Tetris lamp instead.
Link: http://jackm.co.uk/the-simple-proof-of-the-tetris-lamp/
Discussion: https://news.ycombinator.com/item?id=8870479
Link: http://jackm.co.uk/the-simple-proof-of-the-tetris-lamp/
Discussion: https://news.ycombinator.com/item?id=8870479
I was shown a similar problem (I believe it was originally a Putnam question) which asked if you could cover a chessboard missing one corner with L-shaped triominos, and then asked to prove whether or not you could always cover a 2nx2n board with one corner missing. Neat problem
Indeed (spoiler!) an easy, but nice induction proof, which easily can be turned into an algorithm.
"here's another interview brain teaser for sociopathic managers"
Rather a simple exercise for a first-semester exercise sheet.
Only if you already know the trick to attack this class of problems.
No, it's really that easy: Clearly the complete coverings with dominoes biject to matchings of the graph with vertices = squares and edges defined by 2-sets of squares sharing an edge on the chessboard. Clearly this graph is bipartite, but the two elements of the bipartision are of different cardinality. Thus no perfect matching can exist.
Really? Just by looking at the figure, bruteforcing for a couple of minutes and then noticing a solution which gives the same problem but 6x6 instead of 8x8, I was able to solve this without even pen and paper. Unless you're talking about an extremely abstract class of problems, I don't think that statement is true.
We solved this as a high school homework problem.
I wonder if it can be proved without resorting to looking at classes of cases (removed squares right next to each other, removed squares with two squares between them, etc)?