Problem 6 · Character fusions
The invariant $r + 2b$ never changes: the ending is forced.
Integer answer, at most 4 digitsIn a video game there are 95 red characters and 19 blue ones. They can fuse as follows:
- Two red characters fuse into one blue.
- Two blue characters fuse into four reds.
- One of each colour fuses into three reds.
If at the end of the game there are 100 characters in total, what is the minimum number of blue ones? And the maximum? (Answer with the sum of the maximum and the minimum.)
Copa Cangur · SCM
Hard
Closed answer
Reasoned solution
Key idea: look for a quantity that no fusion changes: $r + 2b$ works.
Red+red: $r$ drops $2$, $b$ rises $1$: $\Delta = -2+2 = 0$. Blue+blue: $b$ drops $2$, $r$ rises $4$: $\Delta = 4-4 = 0$. Mixed: $r$ rises $2$ net, $b$ drops $1$: $\Delta = 2-2 = 0$.
$$r + 2b = 95 + 2 \cdot 19 = 133 \quad \text{always}.$$
At the end, $r + b = 100$ and $r + 2b = 133$: subtracting, $b = 33$ forced. Minimum $=$ maximum $= 33$.
$$33 + 33 = 66.$$
Answer: 66