7.4. Find all solutions to the puzzle COW + COW = MILK. Different letters correspond to different digits, the same letters correspond to the same digits

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7.4. Find all solutions to the puzzle COW + COW = MILK. Different letters correspond to different digits, the same letters correspond to the same digits
Answer. Two solutions $(302015+302015=604030$ and $304015+304015=608030)$.

Solution. The equality in the hundreds place could only have occurred in two cases: $0+0=0$, that is, $\mathrm{O}=0$, which could only happen if there were no carries from the tens to the hundreds; and if $\mathrm{O}=9$ (in the case of a carry of one from the tens to the hundreds). However, the sum $\mathrm{A}+\mathrm{A}$ ends in $\mathrm{O}$, so $\mathrm{O}$ is an even digit, meaning $\mathrm{O}=0$ and then $\mathrm{A}=5$.

Next, there is no carry in $\mathrm{K} + \mathrm{K}$, $\mathrm{P}+\mathrm{P}$, or $\mathrm{B} + \mathrm{B}$ (the addends and the sum are six-digit numbers and there are no corresponding carries), so all these digits are no more than 4 (and non-zero: $\mathrm{O}=0$). Therefore, $\mathrm{K}=3$, since $\mathrm{B}+\mathrm{B}+1=\mathrm{K}$. From this, $\mathrm{B}=1$. There are two options left for the digit $\mathrm{P}$, and both work.

Comment. Proved that $\mathrm{O}=0-2$ points.

Found the digits A, B, K - 2 points.

Found only one variant of the digit $\mathbf{P}-1$ point.

Correctly obtained only one of the two answers without justification - 1 point.

Correctly obtained both answers without justification - 3 points.