On Friday I set this puzzle….

I am thinking of four numbers. The mode is 1, the median is 2 and the mean is 3. What are the numbers?

If you have not tried to solve it, have a go now. For everyone else the answer is after the break….

The numbers are 1,1, 3 and 7. That way, the mode (most common) =1, the mean (average) is 3 and the median (the middle number when ordered) = 2. Did you solve it? Any other answers?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.


  1. I also had the answer of 1, 2, 3 and 4. If you write out a puzzle with those four numbers in the question, then you must be thinking about them.

  2. The median value as far as I know should be the one with as many values below it as above it. Unless I am missing something crucial, this does not apply to 3 in that answer – even more serious is that the median was given as 2 in the question and 3 in the answer, something doesn’t quite mesh

    1. Indeed that is the definition of the median. But with 4 numbers, we have an even number of numbers – so there is no middle number. Instead there are 2 middle numbers. In this case, the definition of the median is the average of these 2 middle numbers. And the average of 1 and 3 is 2.

      Richard’s answer is correct and the only answer.

  3. The median is the number in the middle if there is a middle number. In this case, there is no middle number, so it is the average of the two middle numbers. The middle numbers of 1, 1, 3, 7, are 1 & 3. The average of 1 & 3 is 2, so the median is 2.

    1. Nark: No, if there’s an even number of items in the list, the median is the one following the midpoint. You never use a number not in the list as the median. So, with the numbers listed on Friday, the answer is 1 1 2 8.

    1. No, for 1,1,2,8 the median is 1.5 not 2. The puzzle can be reasoned out as follows: Mode=1 means that there must be at least two 1’s. It can’t be three 1’s because that would make the median 1. So there are two 1’s. Median=2 means that the next number has to be a 3 in order for 1 and 3 to average out to 2. That leaves the last number to be 7 since Mean=3 means that the total of the four numbers has to be 12.

  4. You are probably right. It seems the definition of median is not clear (to me). “The median is the middle value in the list of numbers” which would imply the median is actual part of the list of numbers.

  5. The median is when you arrange the numbers in order and then (in the case of an odd number of numbers) choose the one in the middle; or (in the case of an even number of numbers) average the two numbers in the middle.

  6. Heh – both explanations of “median” when there is an even number of numbers say to “take the average”….what “average” would that be then? Mode? Median? Mean?

  7. I had to check up the median for an even number of items before I got the answer. I didn’t know you had to take the mean of the two middle numbers if there were two.

  8. Must admith it would have been nicer if Richard had chosen an odd number of numbers in order not to have an argument ofer the definition of median. But Richard being Richard probably wanted to watch us squirm as usual.

  9. I love the way that many people got the answer “instantly” on Friday, but the reality was that many people didn’t actually get it at all.

    1. I for one will freely admit that I didn’t get it – but then I didn’t say on Friday I did either so maybe it doesn’t count

  10. I learned relatively recently that the median of an even number of numbers is the number half-way between the two numbers in the middle. So, I did get the right answer, relatively quickly.
    Reasoning went like this. Start with 4 blanks. The mode is 1, so we have to have at least two 1’s. Put them in the first two blanks The median is 2, so we have to choose the third number so that its average with 1 is 2. That would be 3, which goes in the third blank. The mean is 3 so the sum of all 4 numbers must be 12. We already have a sum of 5 so the fourth blank must be 7.
    Result :1, 1, 3, 7.

  11. I had forgotten the rule for figuring the median on an even list, and this puzzler forced me to look it up. After that it was easy. Feels good to do some learnin’ and stuff. However, there is clearly a conspiracy afoot. This nugget of surreptitious math education means Richard must be in cahoots with the Math Educators Assistance League (MEAL). Drat. Now I’m hungry.

  12. This is one of the easier ones.
    Four numbers:
    A, B, C, D
    Mode of 1 means at least 2 must be equal to 1
    1, 1, C, D
    Median of 2 means that, assuming C>1 /\ D>C, (1+C)/2 = 2, meaning C = 3
    1, 1, 3, D
    We are left with 2 (tautological) constraints: D>C, (1 + 1 + 3 + D)/4 = 3, which when solved for D yields 7.

  13. So wonderful! I love the earthy & classic tones with the wedding! I’m so stealing the succulent idea. Attractive bride, handsome hubby & bridal party. Astounding footage as often Tammy!

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