9 Nov 2013

Houses in a Row: Professor Pailyn's Mathematics Quest PMQ44

“Look, I’ve invented a problem that I can’t figure out!” Alice sounded exasperated. “I thought it was going to be easy, but it isn’t. Imagine four houses placed on a map and each connected to all the others with a straight road. Now, I was wondering whether the length of every road could be a whole number. That bit is easy! So then I thought about whether the length of every road could be a different whole number. That’s when I got stuck!” Alice slumped in her chair, just to illustrate her defeat.

“Your problem does, indeed, have solutions.” Soothed Professor Pailyn, “But the mathematics needed to find them may take a while to go through. It just means learning some new topics that you probably haven’t yet done in school. We can do it, but your problem has made me think of a similar situation.” The Professor took some paper and sat down next to Alice.

“Imagine your four houses are connected by a straight line and, as you said, you want the distance between every pair of houses to be a different whole number. Now, try to find the solution so that the longest distance is as small as possible.”

“But that’s too easy! Just lay them out with gaps of one, two and three!”

“Erm... not so fast! If you do that, then the distance between these two houses is 3... kilometres, say, and so is the distance between these other two.”

“Oh yes, of course – how foolish of me. Let me think...” Alice started thinking...


“That was nearly too easy!”

“OK, but they get harder. Now try this with 5 houses. You can label the houses A, B, C, D and E. You want every distance between pairs of houses to be a different whole number. What is the smallest value for the distance A to E?”

“OK, here goes...” Alice was on a roll.

“When you’ve solved that one, try it with 7 houses.” Professor Pailyn said on his way out into the garden to do a bit of pruning.


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