Middle Digit Mean
Weekly Problem 16 - 2016
How many three digit numbers have the property that the middle digit is the mean of the other two digits?
How many three digit numbers have the property that the middle digit is the mean of the other two digits?
Problem
How many three digit numbers have the property that the middle digit is the mean of the other two digits?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
If the three digit number is "$abc$", then we need $b = \frac{a+c}{2}$.
As $a$ and $c$ are chosen to be between $0$ and $9$, $b$ will be certainly, as it is the mean. This means we only need to check which cases make $b$ an integer.
$b$ is an integer exactly when $a+c$ is even.
If $a$ is even, $c$ needs to be even also. If $a$ is odd, $c$ needs to be odd also.
If $a$ and $c$ are both even, $a$ can be any of $2$, $4$, $6$ and $8$, so four options. $c$ can be any of $0$, $2$, $4$, $6$ and $8$, so five options. This means there are $4 \times 5 = 20$ combinations.
If $a$ and $c$ are both odd, they can be any of $1$, $3$, $5$, $7$ and $9$. This gives $5$ options for each, so there are $5 \times 5 = 25$ combinations.
Thus there are $20 + 25 = 45$ three digit numbers with this property.