In this example the corners of the chessboard are both black, so if you cut out these corners you cannot fill the board with dominoes. Therefore, if you do not have an equal number of white and black squares available on your customized chess board, you cannot fill it with dominoes. It is impossible to lay a domino on a chess board otherwise. Took me a little while to figure this one out…Īny domino is going to take up exactly one white square, and one black square.
Success is determined by those who prove the impossible, impossible. Success is determined by those who prove the impossible, possible. In the subject line to Please include the solution to the puzzle and indicate whether or not the puzzle is original. To submit a puzzle, send an email message with “NEW LAB PUZZLE” Someone who comes up with an especially interesting answer or to someone who proposes a sufficiently intriguing puzzle for Lab readers to solve. The weekly prize, a copy of “Mathematical Puzzles of Sam Loyd” (edited by Martin Gardner), will go to (Is there even a theme? Hmm, that’s odd, I didn’t put it in there…).Īs usual, you can submit answers as comments here (but we won’t publish any until tomorrow). Can you come up with more sentences like it, preferably topical?Īs always, any literary efforts or poems related to the theme of today’s puzzle is welcome. The object of the game is to leave only one peg standing at the end.Ī) Is it possible to reach the single-peg endpoint with the starting position illustrated here?ī) If the blank square were in the center square, with all the other squares filled in with pegs, then could youįor our word challenge today, consider the following sentence:Īll the words have alternating consonants and vowels ( y is counted as a vowel when it makes a vowel sound). A peg can jump horizontally or vertically (but not diagonally) over another into an empty space immediately beyond it. At right is a picture of a classic peg-jumping game board. Can the resulting board be completely tiled using 31 of the 2×1 dominoes no matter what theĢ. If you remove two random squares - one white and one black - from a regular 8×8 chessboard. Is it possible to use 31 of these dominoes to tile all 62 squares? Suppose you have a set of 2×1 dominoes, each of which covers two squares that are adjacent vertically or horizontally. We begin with the classic cornerless chess board:Ī) When you cut off the two diagonally opposite corner squares of a regular 8×8 chessboard, as shown above, 62 squares Who knows, it might save you from getting a lot of tender lumps on the noggin.ġ. Whether a solution is possible or impossible. Today and next Monday I’ll give you challenges where the emphasis is not on solving the puzzle but determining Who knows?įortunately, sometimes there are ways to tell if a problem is impossible, just by reasoning about it. Hand, if they aren’t, maybe some mathematicians gave up too soon.
Are they impossible? If they are, then some mathematicians may be butting their heads against them in vain. On the other hand, there are other mathematical puzzles such as Riemann’s Conjecture or P=NP that have been unsolved since they were proposed, the former, one hundred and fifty years ago. Known as Fermat’s last theorem, was unsolved for over 350 years, and many mathematicians thought it was impossible. Nature, however, gives us no such guarantees. In solving puzzles, there is an implicit trust that the solver has to place in the puzzle giver that there is really a solution to the puzzle. Can you precisely cover this board with dominoes that each occupy two adjacent squares?
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