Enigma 295: The max-multiple game
16 July 2015
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From New Scientist #1443, 14th February 1985 [link]
This is a game between you and the Angel. It starts with the natural numbers from 1 to N, written in a row. You and the Angel play alternately, you first. The rules are:
(a) You take any number you choose (subject to D below) from those remaining in the row, and delete it from the row.
(b) The Angel deletes from the numbers remaining in the row all these which are multiples of the number you just took.
(c) Go to (a).
(d) You can never take a number which has no multiple remaining in the row; that is, your take must permit the Angel in his turn to delete at least one number.
The games stops when you can legally take no more numbers, and you want the sum S of all the numbers you have take to be as large as possible.
The picture records a game with N=9 and S=8. You could have done better. Now try with N=35. How large can you make S?
Also, today is (Spoiler Alert!) Cheryl’s Birthday!