Game Teachers' Notes Solution

# Factors and Multiples Game

Ian's thoughts:
I figure that it is not worth using prime numbers greater than 25, or their multiples.
That rules out Prime numbers
p: 29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97
2p: 58, 62, 74, 82, 86, 94
3p: 87, 93
That rules out 24 numbers.
So I suspect that you can't beat the score 76. Although I should add that in my 66 I DID use multiples of 29 right at the end because I had nothing else on.
What would be useful to get a high score would be if in the top right corner of each number N there was a digit d that was continually updated which indicated how many options you have after pressing N. A good strategy would be to try to clear out those numbers N with low d early on. (The numbers p, 2p, and 3p that I list above would have low d, but it would still be better to ignore those numbers.)
I suggest this for an opening (multiples of 13): 39, 78, 26, 52, 13, 65, 5
That gets rid of most of the troublesome multiples of 13 early on. It misses out 91 though, which you are unlikely to later recover.
I think it might be difficult to find the highest possible score without writing some code.
Further thoughts:
In a very loose sense, there are three categories of numbers:
[1] Numbers which you should ignore and not select at all. i.e. large prime numbers.
[2] Numbers which you should select in groups. e.g., multiples of 13. Get rid of them all in a row.
[3] Numbers which are easy to get rid of, and which should be used for moving between the groups.
For example, consider the number 5. You want to get rid of the group of multiples of 13, and the group of multiples of 11. You go through the multiples of 13 successively, and finish with 65=5*13.
Then you select the number 5. Then you select 55=5*11.
Then you move into the multiples of 11.
This is certainly the right sort of procedure.
One thing that might help the process is if a graph was produced. That is, each number was connected by a line to all numbers which are either a factor or multiple. Then you would see the groups all connected together, and the idea of moving between them would seem more concrete.
(They would be like little towns connected by little roads, then the useful numbers like 5 would connect towns together.)
Then you would see that 55 has lines only to 5 and 11 (and 1), which makes it a hard number to use.
I reckon I can prove that a user can't score higher than 82. I might be able to knock that down a bit, but I don't think I can prove that 77 is optimal. I strongly suspect that 77 or 78 is optimal, with an outside chance of 79.
Here are Ian's best attempts so far: 72 , 75 and 77 numbers selected