27 Nov 2013

A Tangled Peg-Board: Professor Pailyn's Mathematical Quest PMQ46

Alice was playing around with a peg-board and a jar of elastic bands. She was trying to think up a devilish problem to give Brenda when she finally turned up. Alice had made a real tangled mess and was annoyed with herself for then having to take all the rubber bands off... and then one peg snapped off! She stared at the board, as if willing the peg to jump right back into its rightful place; but it didn't, of course.

However, this now meant that Alice could play with a smaller board. She laid out a perimeter so that she was left with a board of 6 by 6 pins (as shown in the diagram). She had thought of how many elastic bands she'd need if she joined together every pair of pins that were a whole number distance apart, but that had ended up badly. So then she had another idea: join together every pair of pins that are a prime number distance apart. That sounded better! Assuming, of course, that the distance between adjacent pins was a unit length.

So, how many elastic bands will Alice need this time? Do you think she'll have enough of them?!

Note, the diagram shows some examples. Also note that the bands may well overlap each other, but so long as they join two different points that's OK. For example, the horizontal line shown joining 4 pegs contains 3 rubber bands, whereas the vertical line has just 1 band.

Now, where's that Brenda? She's late!


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