$\Large \square \, \square \times \square = \square \, \square \, \square$

In each $\square$ , we use a distinct digit: 1, 2, 3, 4, 5, 6.

What is the right hand side?

The answer is 162.

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The single digit in $\square$ can't be $1,5,6$ because If you multiply a two-digit number by 1, you get the same number. If you multiply a two-digit number by 6, with last digit $1,2,3,4,5$ , you'll get a number ending in 6,2,8,4,0 and this implies that this case is impossible . If you multiply a two-digit number by 5, the last digit will end in 0, 5,. So this case is impossible too, (why?).

Therefore, the single digit in $\square$ is $2,3$ or $4$ . Playing a little bit, you'll get $54 \times 3 = 162$ .

Final review.-We just only have the digits $1,2,3,4,5,6$ and we can just only use it once. If you multiply a two-digit number $A$ by $\boxed{2}$ , to get a 3-digit number, $A$ must start with 5 or 6. But, then what happens?... ( $65 \times 2 = 130$ , uff, you/we almost get it :)... )If you multiply a two-digit number $A$ by $\boxed{4}$ , to get a 3-digit number, $A$ must start with 3,5 or 6. But, then what happens?... If $A$ start at 3, what happens?... If $A$ start with 5 , you'll get a three digit number starting at 2, and if you multiply a number by 4 with last digit $1,2,3,5,6$ you'll get a number ending in 4,8, 2, 0, 4. For example, ( $53 \times 4 = 212$ ,uff, you/we almost get it :)...). If $A$ start at 6 , you'll get a three digit number starting at 2. For example, ( $63 \times 4 = 252$ ,uff, you almost get it :)...).

So, we only can use $\boxed{3}$ like single digit in $\square$ . If you multiply a two-digit number $A$ by 3, to get a 3-digit number, $A$ must start at 4,5 or 6. A two digit number ending in $1,2,4,5,6$ multiplied by 3, you'll get a number ending in 3,6,2,5,8. This implies that the two digit number can't start at 4 (why?) (Hint: Use hit, hit and trial... $42 \times 3 = 126$ uff, you/we almost get it).The two digit number can't start at 6, either, (Why?) (Hint: this time don't use hit and trial) . Therefore, The only possibility is $54 \times 3 = 162$