I came across the following question in an mathematics journal awhile
ago, and it has been puzzling me for quite some time now!

Any solutions would be welcome...

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"Two players, A and B, take turns choosing a number between 1 and 10
(inclusive).  A goes first. The cumulative total of all the numbers
chosen is calculated as the game progresses.  The player whose choice
of a number takes the total to exactly 100 is the winner."
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* How would one use backward induction to find each player’s best
strategy? *

If Player A chooses a number that increments the total to between 90
and 99, then Player B can choose a number such that B wins.  The game
then becomes who can get the count to 89.  If A can get 89, B must
select a number, yet can't win, and A can then win on the next round. 
Continuing this theme, the goal becomes to be the player to select to
78.  If A scores 78, B chooses any number 1 through 10, and A then
chooses the number to bring them to 89.  Going backwards, the ideal
numbers become 89, 78, 67, 56, 45, 34, 23, 12, and 1.  Player A should
select 1 as the first number, and then choose accordingly to reach
each of the named scores.  If A does so, B should never win.