First let's state the game:
Game: Matin and Amo met each other in a gym and decided to play a game. The gym has
long enough horizontal metal shaft and exactly one weight of each radii $1, 2,\dots, 100$ units. They
take turns alternatively, starting with Amo, picking up a weight and inserting it from either left
of right head of the shaft. After all weights have been inserted, Matin stands by the left side of
the shaft and Amo stands by its right; among them the one who can see more weights wins; if
they see equal number of weight then it is a draw. By seeing a weight we mean that no weight
with bigger radius exist between the observer and that weight. Considering that they both are
intelligent mathematicians, determine the result of this game.
Here are my results:
- Matin has a draw strategy:
If Amo starts with inserting the weight with radius 100, Matin inserts 99 from his end, after this move Matin just needs to insert the largest weight from his end; in this manner any weight inserted by him will be visible by him and not Amo. So Matin will see 51 weights and Amo will see 50 weights. But if Amo start with a weight with radius less than 100 Matin puts the weight with radius 100 at Amo's end and continues like the other case. - Playing 99 when 100 is not played yet will cause the rival to have a draw strategy:
This is actually quiet easy the rival will insert 100 from not his but the other end of shaft. The rest is similar to the above when 100 is inserted. - Playing 98 before 99 and 100 will cause the rival to have a draw strategy:
Again the other player does the same as above it is not hard to see it works.