Finding the Average of 3-Digit Numbers

[Ja4Coltrane]

Homework Statement

How many three digit numbers are there that the middle digit is the average of the first and last digit.

The Attempt at a Solution

I started with 9 because of 111, 222,...999
then I added 7 for 123, 234,...789
then 5 for 135, 246,...579
3 for 147, 258, 369
1 for 159
that gives 24 but I doubled it for 123 and 321
that's 48 but I then subtracted 9 for 111=111
that leaves 39 which is not one of the possible correct answers.

 

[HallsofIvy]

Does 011 or 000 count as a "three digit number"? I'm going to assume it does not.

In order that the average of two numbers, (a+b)/2, be an integer a+ b must be even which means that either the two numbers are both even or that they are both odd. The first digit can be any of 1 through 9. If the first digit is odd, the last digit can be any of the 5 odd digits. If the first digit is even, the last digit can be any of the 5 even digits (we can count 0 now). That is, there are 5(9)= 45 choices for the first and last digits. But the requirement that the middle digit be the average of the first and last means that the middle digit is "given" by the first and last- there are no other choices. There are 45 possible 3 digit numbers such that the middle digit is the average of the first and last digits.

If 000, 00a, 0ab are allowed, then there are 10 choices for the first digit and, still, 5 choices for the last digit for each of those. There are 10(5)= 50 such numbers.

 

[Architeuthis Dux]

I would approach this problem in a more systematic way. Firstly, I would define the problem clearly by stating that we are looking for three-digit numbers where the middle digit is the average of the first and last digit. Then, I would list out all the possible combinations of three-digit numbers that fit this criteria, starting with the smallest number (101) and ending with the largest (999). This would give me a total of 243 numbers.

Next, I would check each number to see if the middle digit is indeed the average of the first and last digit. For example, for the number 123, the middle digit is 2, which is the average of 1 and 3. I would keep track of all the numbers that meet this criteria.

Once I have gone through all 243 numbers, I would have a final list of the three-digit numbers where the middle digit is the average of the first and last digit. I would then count the total number of numbers on this list, which would give me the correct answer.

In conclusion, the correct answer to this problem would be 39, as stated by the attempt at a solution. However, a more systematic approach would be necessary to arrive at this answer accurately.