At the same time, Black widow goes to Vormir to get the soul stone. She is stopped by Red skull who says that she can obtain the stone only if she plays a logic game with him. He starts explaining the rules of the game as follows: There are two piles of pebbles, each pile less than equal to 15 pebbles. The two players take turns picking up pebbles.
1. You can pick up any or all the pebbles in one pile, or
2. You can pick up pebbles from both piles, but only if you take the same number from each pile.
The player who takes the last pebble wins. Winning positions include (1, 0) (Meaning 1 pebble in the first pile, none in the second) and (n, n). You pick up, respectively, 1 pebble (rule 1) and 2w pebbles (rule 2). (1, 2) is a losing position: But though (1, 2) loses, not only (1, 0) and (1, 1) but any other (1, n) wins. A player simply picks up (n — 2) pebbles from the second pile, leaving (1,2). There are many other losing positions other than (1,2). Help her find all such losing positions. What is the total sum of the differences of the two numbers of all the distinct losing pairs?