Newsletter

Chris, Scotland

Friday, August 1, 2025

"First, there are one digit numbers whose digits are, by definition, the mean of their digits… so -9,-8,-7,….7,8,9.

Two digit numbers have a mean at most 9 so that’s a dead end.

We’re going to have to hunt for decimals if we’re to uncover further answers…

If they’re two digits:

4.5 does the business (9/2=4.5)

The only other ones I’ve come up with have a fixed number of declared zeroes at the end like:

1.500 = 6/4
2.250 = 9/4
3.750 = 15/4
1.8000 = 9/5
1.1250000 = 9/8
1.687500000000000 = 27/16


Those seem like cheating as obviously there are more zeroes following the zeroes I’ve stuck on the end. And maybe your readers will uncover more!

Thanks for a fun puzzle! I enjoyed finding solutions "

Mala, New Zealand

Saturday, August 2, 2025

"I think the only answer to this is the number 4.5 as it is equal to the mean of 4 and 5 (which is 9/2 or 4.5).


"

Leonard, United States

Saturday, August 2, 2025

"Here’s what I have so far.

Obviously, all ten single digits work: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. In these cases, the number also equals the median and mode of its digits.

I believe the standard answer is: 4.5 = (4 + 5) / 2 .

How about: 20 = 1 = (2 + 0) / 2, or 1! = 1, or 2! = 2.

Also: 1 / 0.5 = 2 = (1 + 0 + 5) / 3 .

10 / 5 = 2 = (1 + 0 + 5) / 3 . This one “pairs” with 1 / 0.5.
24 / 6 = 4 = (2 + 4 + 6) / 3
35 / 7 = 5 = (3 + 5 + 7) / 3
54 / 9 = 18 / 3 = (5 + 4 + 9) / 3

Or, stretching the rules a bit: 2.250 = (2 + 2 + 5 + 0) / 4 = 2.25, and 3.750 = (3 + 7 + 5 + 0) / 4 = 3.75 .

And, if you bought those: 0.500000000 = (0 + 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0) / 10 = 0.5 .

Still working on this great puzzle. I might be at it all month.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

My final (not complete) set of fractions.

36/12, 56/14, 81/18, 65/20, 63/21, 92/23, 90/24, 90/30

126/42, 162/45, 288/48, 153/51, 286/55, 232/58, 180/60, 288/60

396/66, 315/75, 304/76, 340/85, 493/85, 324/90, 486/90, 558/93

I’m thinking that 8.9 repeating is more of a philosophy question than a mathematical one. I need to review some sequence and series topics I’m sure I knew at one point. "

Rick, United States

Friday, August 8, 2025

"I have had what I perceive is the desired answer to the august 2025 for several days but wanted to explore other possibilities. However, this is taking longer than I planned, so I am sending you my solution before I forget. Hopefully, I will have an update later this month.

What number equals the mean of its digits?

The mean of a set of numbers is calculated as:

Mean=Sum of all numbers/Number of elements

However, the mean is valid for a single number.

So, if the set contains only one number, say, x, then:

Mean=x/1 or x

In this case, the mean is simply the number itself. This is mathematically consistent and valid,

Hence, the numbers 0 through 9 satisfy the solution to the problem.

However, I doubt this was the desired solution, so I looked at alternatives.

Since the mean consists of a set of numbers in the range of 0 to 9, the mean has to be a number in the range from 0 to 9. Therefore, we can eliminate any numbers greater than 9. The only way to get more digits is to look at fractional or decimal numbers. I first considered fractional numbers. I found a couple of solutions, none extremely interesting, but I include them here for completeness:

1 - 0/2, which is equal to 1 and so is its mean.
2 – 04, which is equal to 2 and so is its mean.

Hopefully, I can find more interesting fractions. This may require some Python scripting.

Finally, I decided to investigate decimal numbers. Obviously, 0.0, 0.00, 0.000 , and so on satisfy the solution space. But I had a feeling this still was not the intended answer.

To search for what I perceive was the desired solution, I initially considered decimal numbers with only one number following the decimal place (a number consisting of only two digits). that the resulting mean of a number with only two digits would either be a whole number or a number with five in the decimal place. Since the value of the mean of two digits which add up to an even number would result in a whole number value, we only need to consider numbers like 2.0, 4.0, 6.0, and 8.0. But none of these satisfy the solution space, since their mean is half their value. We need to consider the numbers 2.5, 4.5, 6.5 and 8.5, of only which 4.5 satisfies the requirement that the mean is equal to the number.

So, I submit that 4.5 is the intended solution to the August 2025 puzzle of the month. "