# 1997 PMWC Problems/Problem I2

## Problem

In the multiplication in the image, each letter and each box represent a single digit. Different letters represent different digits but a box can represent any digit. What does the five-digit number $\mathrm{HAPPY}$ stand for? $$\begin{array}{c c c c c}& &\Box & 1 &\Box\\ &\times & & 9 &\Box\\ \hline &\Box &\Box & 9 &\Box\\ \Box &\Box &\Box & 7 &\\ \hline H & A & P & P & Y\end{array}$$

## Solution

Following the rules of multiplication, we see that 9 times the units digit of the three digit number ends in 7, which means that the digit must be a 3. Carrying out the multiplication, we see that the last two digits of the second product are 17, which means that the hundreds digit in the first product must be a 4. We now have

   -13
*9-
___
-49-
--17
_____
HA66Y


The only digit that would work as the units digit of 9- is 7. Therefore we have

   -13
*97
___
-491
--17
_____
HA661


The only multiple of 7 that is two digits and is 7 times a digit is 14. Therefore we have

   213
*97
___
1491
1917
_____
20661