Mystery Matches

Relevant wiki: Logic Puzzles - Intermediate

In order to remove 2 matches from the left, we can opt to remove 2 matches from 1 digit or 1 match from 2 digits each.

For the former case, only the digits $8$ & $9$ can be applicable: $8$ is adjusted to either $2$ , $3$ , or $5$ ; and $9$ to $4$ , as shown below.

Then for the above adjustment, we can calculate the possible outcomes as the following:

$67\times 29 = 1943$

$67\times 39 = 2613$

$67\times 59 = 3953$

$67\times 84 =5628$

Obviously, none of the products can be obtained by moving only 1 match on the right side. Therefore, the process must be completed by taking out 1 match from 2 digits each, and the possibilities are as followed:

Now we will choose a pair of digits to have the 2 matches removed, and there are $6$ possible pairs.

Starting from the pair 6 & 7 , there is only one possible adjustment:

$51\times 89 = 4539$ .

This can't be done by moving one match within the right side.

Then for the pair 6 & 8 , there are two possibilities:

$57\times 69 = 3933$

$57\times 99 = 5643$

They are not correlating to the rule, either.

Then for the pair 6 & 9 , there are two possibilities:

$57\times 83 = 4731$

$57\times 85 = 4845$

Again, these are not the solution.

Now moving forward to the pair 7 & 8 , there are two possibilities:

$61\times 69 = 4209$

$61\times 99 = 6039$

Still, they are not our desired outcomes.

Then for the pair 7 & 9 , there are two possibilities:

$61\times 83 = 5063$

$61\times 83 = 5185$

Finally, we got $5063$ which can be obtained by moving one match within the digit $9$ in the original product $5963$ . We will check if there are other solutions.

Last but not least, for the pair 8 & 9 , there are four possibilities:

$67\times 63 = 4221$

$67\times 65 = 4355$

$67\times 93 = 6231$

$67\times 95 = 6365$

At last, we have checked that no other products can be formed by moving only one match on the right. Thus, the answer is unique.

As a result, when we remove the matches from $7$ & $9$ , forming $1$ & $3$ respectively, on the left, the equation can be made true by moving the match of $9$ to be $0$ on the right side as shown.

The new equation will then be $61\times 83 = 5063$ .