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Which digit does X represent in this column sum?
Problem
Each letter in this sum represents a different non-zero digit.
Which digit does X represent?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Answer: 7
Using trial and improvement
Ones (units) column: E + E gives S
Tens column: E + E gives E
So E + E must be 10 or more, so that 1 is carried to give a different result in the two columns
Try E = 5
E can't be 5, because should have E in the tens column
Try E = 6
E should be larger
Try E = 8
E should be larger
Try E = 9
That works, so E must be 9, and S must be 8. So we have
And X = 7.
Using algebra
Notice that the ones (units) column has $E + E$ and gives $S$, but in the tens column, $E + E$ gives $E$. So $E + E$ must be $10$ or more, so that $1$ is carried to give a different result in the tens column.
So $E + E = 10 + S$, and $E + E + 1 = 10 + E$.
$E + E + 1 = 10 + E$ means that $E + 1 = 10$, so $E = 9$.
So since $E + E = 10 + S$, $18 = 10 + S$, so $S = 8$.
So we have
And so $X=7$.