{"id":589,"date":"2013-02-02T16:16:30","date_gmt":"2013-02-02T08:16:30","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=589"},"modified":"2024-02-16T21:13:01","modified_gmt":"2024-02-16T13:13:01","slug":"making-tough-puzzles-easy-or-less-tough-anyway","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2013\/02\/making-tough-puzzles-easy-or-less-tough-anyway\/","title":{"rendered":"Making Tough Puzzles Easy (Or Less Tough, Anyway)"},"content":{"rendered":"

My in-laws in Singapore called up, asking for a solution to this puzzle<\/a>. A short while later they called again to say that they’d solved it, but only after a long period of trial and error, asking me if there was a more systematic way.<\/p>\n

<\/p>\n

The puzzle is this : Arrange the numbers 2 to 9 in the grid below, so that the two rows and the two columns each add up to 16.<\/p>\n

\"The<\/a>
The grid for the puzzle<\/figcaption><\/figure>\n

Eventually, we could guess the answer. Is there a way to avoid guessing?<\/p>\n

The answer is, “Yes and No”. You can’t avoid guessing altogether, but with a bit of careful thought, you can drastically reduce the range of guesses than need to be checked.<\/p>\n

Just as for the puzzle of working out the 3×3 magic square<\/a>, it pays to ask what happens if we add certain rows and columns together.\u00a0In this case, let’s add up the (unknown) numbers in the top and bottom rows.<\/p>\n

\"Can<\/a>
Can we find out what L and R add up to?<\/figcaption><\/figure>\n

Well, the puzzle says that each row and column adds up to 16, so these rows together add up to 32. Gosh, that’s easy. Let’s ask something harder : what’s left over ? What’s the sum of the cells on the left and right marked L and R ?<\/p>\n

The puzzle says that we have to arrange the numbers 2 to 9 in the grid. 2+3+4+5+6+7+8+9 adds up to 44. The two rows add up to 32, so there’s 12 left. The cells L and R add up to 12.<\/p>\n

There’s only a few pairs of numbers from 2 to 9 that add up to 12, namely there’s 3+9, 4+8 or 5+7. We can ignore 6+6, since we’re only supposed to use each number once. So, if we were trying to guess the values of L and R, there’s only three reasonable guesses (instead of 28 if we hadn’t thought of this trick).<\/p>\n

The good news continues, since the same logic that tells us the sum of the column centers also tells us the sum of the row centers.<\/p>\n

\"T<\/a>
T and B also add up to 12<\/figcaption><\/figure>\n

If you add up the left and right columns, that adds to 32. Adding on T and B as well has to make up the total 2+3+4+5+6+7+8+9, just as before, so T+B must be 12 as well.<\/p>\n

So, to solve this puzzle, you don’t need to guess everything. You just need to guess which of 3+9, 4+8 or 5+7 make up the left and right cells L and R, and which of 3+9, 4+8 and 5+7 make up the top and bottom cells T and B. There’s really only three possibilities to try (since the puzzle is very symmetrical), and then guessing a single corner gives you all the rest. Make your guesses systematically, and you’ll have quickly found the one and only solution to this puzzle.<\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

My in-laws in Singapore called up, asking for a solution to this puzzle. A short while later they called again to say that they’d solved it, but only after a long period of trial and error, asking me if there was a more systematic way.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[277,439,440,7,397,441,438],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/589"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=589"}],"version-history":[{"count":3,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/589\/revisions"}],"predecessor-version":[{"id":1320,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/589\/revisions\/1320"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=589"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=589"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=589"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}