πŸ”΅πŸ”΄ Last Ball

Balls are repeatedly removed from a bag according to a simple rule. The individual draws are randomβ€”but the color of the final ball is actually predetermined.

Current bag
20 blue Β· 14 red
Guaranteed final: πŸ”΅ Blue

Click a specific pair to force that case, or use a random draw.

Pick two balls to begin.
The three possible transformations
πŸ”΅ πŸ”΅
β†’
πŸ”΅
Ξ”Blue = βˆ’1, Ξ”Red = 0
πŸ”΄ πŸ”΄
β†’
πŸ”΅
Ξ”Blue = +1, Ξ”Red = βˆ’2
πŸ”΅ πŸ”΄
β†’
πŸ”΄
Ξ”Blue = βˆ’1, Ξ”Red = 0
πŸ”’ Invariant: red parity
14 red β†’ EVEN

Look only at the number of red balls. Every possible move changes the red count by either 0 or βˆ’2.

Therefore:

Red count mod 2 never changes

R0 mod 2 = 0 = R1 mod 2 = … = Rfinal mod 2
Try different starting bags
Why randomness doesn't matter
0 moves
At the beginning
14 red = even
Every operation
Red changes by 0 or βˆ’2
When one ball remains
Red must be 0 β†’ πŸ”΅

The key insight

Don't try to predict the sequence of random draws. Instead, find something that cannot change. Here that quantity is the parity of the number of red balls. If red starts even, the final ball must be blue. If red starts odd, the final ball must be red.