Worms

Here are three solutions to this problem from Anna, Oliver and Lucy.

From Anna we had the following which was very well presented:

I tried out two numbers and compared them. The numbers were 34 and 67.
These were my results:- 34 is the head of the worm

34 + 24 + 23 + 33 = 114	34 + 24 + 25 + 35 = 118
34 + 44 + 33 + 43 = 154	34 + 35 + 45 + 44 = 158
34 + 24 + 14 + 4 = 76	34 + 35 + 36 + 37 = 142
34 + 44 + 54 + 64 = 196	34 + 33 + 32 + 31 = 130
34 + 24 + 14 + 15 = 87	34 + 35 + 36 + 46 = 151
34 + 44 + 54 + 53 = 185	34 + 33 + 32 + 22 = 121

I found 18 more but the highest score that I found was 196, and it was a vertical worm, crossing through 34, 44, 54 and 64. These are some of the worms where the head starts at 67:

67 + 77 + 87 + 97 = 328	67 + 57 + 47 + 37 = 208
67 + 68 + 69 + 70 = 274	67 + 66 + 65 + 64 = 262
67 + 57 + 56 + 66 = 246	67 + 68 + 58 + 59 = 250

I found 18 more of these, but the highest score was 328.

I wonder what would happen if I could use any numbers in the square?
I wonder what would happen if I could draw the worm diagonally? Would the score be higher?
I wonder what would happen if there were more numbers in the square?
I wonder what would happen if the square was a triangle or a different shape? I wonder?

Well Anna, thank you so much for your thoughts. I liked the picture of you thinking these thoughts.

Stephen in the same school [West Flegg in Yarmouth] did some similar work and his thoughts were:-

I wonder what would happen if there were two worms and not one?

Oliver, from a little village near Somerton in Somerset,drew the starting field of 100 in the way suggested.

Then he drew, quite systematically the following worms and showed their additions with the head on 43.

Well done Oliver!

Lucy, from Wells in Somerset, started off with this different arrangement of numbers:

She then coloured in the worms as solutions and I've used the same colours to colour in the ones that she drew together with the totals.

She then numbered the square in a very interesting way. See if you can see what she has done and then perhaps go one to do a very different one for yourself.

Unfortunately she did not have time to do very many, but I got caught up with this idea so I did the last twenty showing the kind of shape I was using. Then I began to notice things!!!