What you don't understand is that for any fix total amount of minutes you want to play him (this is any given constant you want), the goal is to maximize the resting time in between playing minutes. Resting time before the first playing interval or after the last playing interval are not considered because he is not resting from a previous playing interval or saving energy for the next playing interval. At the beginning of the game he is already rested, so he doesn't need to rest. At the end of the game, even if he is dead tired it doesn't matter, because he will have time to recover for the next game. The critical issue here is the resting time between playing intervals. You want to maximize this in order to maximize player productivity when he is on the field during the game. If a player is too tired after a particular playing interval, and the resting time is not sufficient, the player will decrease productivity in the next playing interval. Let me reemphasize that the total amount of minutes played should remain constant for whatever scheduling policy you enforce.
For example, let's assume you want to play him for 24 minutes. This is the total time you want to play him, and this is a given constant. What is the worst and optimal scheduling policy that minimizes and maximizes resting time in between playing intervals?
Assuming you don't have constraints as to the maximum playing interval (allow me this to make the explanation simpler), the worst scheduling would be to play him 3rd and 4th quarter of the game. He would rest the 1st and 2nd quarter, and then play continuously for 24 minutes (any two consecutive quarters would do, not considering timeouts or half period resting). This policy would give him 0 minutes of resting time between playing intervals, because he would play just one big time interval. The best sheduling policy would be to play him the entire 1st quarter, rest him the 2nd and 3rd quarter, and then playing him the last quarter. The total amount of time played would be 24 minutes (same as the previous case), BUT the resting time between playing intervals would be maximized to 24 minutes (instead of 0 minutes as the worst case above).
Of course, there are other constraints like the maximum playing interval y, critical parts of the game, crunch time etc. The strategy that yields maximum benefits still consists of playing the player at the beginning and end of the game, not only in terms of maximizing resting time between playing periods but also to increase the flexibility of a dynamic scheduling policy determined by a coach (you can mathematically prove the last statement, but not in this forum). This is a basic constraint scheduling problem.

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