John was doing his math problems like a good child one day until he looked at the clock to see if it was time to sleep. To his dismay, the blasted thing had become another math problem! But this one was particularly fiendish! He couldn't make heads or tails of it! He toiled and toiled until he could bear it no longer and smacked the thing with a shoe. It fell on the ground with am mighty crash, but nothing broke except a single one of its numbers. He could tell what color it was, and the slot it was in was still together, but the identity of the number was all but lost.

What number does the question mark represent, and if applicable, what are the rule for the colors? (numbers and circles)

Apologies for any errors in formatting or logic, this is my very first!

The clock

Hint 1:

The fact that the numbers are arranged on a clock seems important...

Starting from basic observations would be useful here.

Hint 2:

This is a tricky one for sure, make sure to check your bases before working on it some more

  • $\begingroup$ Is the image visible? It's not visible on my screen $\endgroup$ Commented May 3, 2020 at 16:28
  • $\begingroup$ Yes, it's visible now. Do you have Imgur blocked, by any chance? $\endgroup$ Commented May 3, 2020 at 16:29
  • $\begingroup$ I definitely don't, strange $\endgroup$ Commented May 3, 2020 at 16:40
  • $\begingroup$ Very weird, turns out ublock origin was blocking imgur $\endgroup$ Commented May 3, 2020 at 21:06

2 Answers 2


Partial answer, mostly observations but lots of them. Might be useful as a springboard if someone else can spot a unifying pattern between all of these.


  • Colours for circles are

    yellow (2,8,12), green (1,5,11), red (3,9,10), blue (4,6,7).

  • Colours for numbers are

    yellow (6,8,9,11), red (1,3,4,12), blue (2,5,7,10).

So we notice that

each combination of circle-colour and number-colour appears exactly once. There are 4 possible circle-colours and 3 possible number-colours, and $4\times3=12$.

More observations, focusing now on the same-colour patterns:

  • The numbers in green circles are all $11$. One of them is at the 11 position on the clock.

  • The numbers in yellow circles are $50$, $100$, and unknown - maybe they're all multiples of a common factor? The number $100$ is a multiple of the sum of the clock positions of the other two ($12+8=20$).

  • The numbers in red circles are $3$, $56$, $122$, but they don't have any common factor except two of them sharing the factor 2. One of them is at the 3 position on the clock.

  • The numbers in blue circles are $13$, $16$, $231$, but they don't have any common factor. The number $231$ is a multiple of the sum of the clock positions of the other two ($4+7=11$).


  • The blue numbers are $3$, $11$, $16$, $100$.

  • The yellow numbers are $11$, $50$, $56$, $231$.

  • The red numbers are $11$, $13$, $122$, and unknown.

My theory is that the number in each circle depends in a fixed way on

the other numbers with the same colour, and the other numbers in same-coloured circles - or rather, not on the numbers written there, but on their clock positions.


$11$ as a function of $\{5,11\}$ and $\{3,4,12\}$
$100$ as a function of $\{8,12\}$ and $\{5,7,10\}$
$122$ as a function of $\{9,10\}$ and $\{1,4,12\}$
$13$ as a function of $\{6,7\}$ and $\{1,3,12\}$
$11$ as a function of $\{1,11\}$ and $\{2,7,10\}$
$231$ as a function of $\{4,7\}$ and $\{8,9,11\}$
$16$ as a function of $\{4,6\}$ and $\{2,5,10\}$
$50$ as a function of $\{2,12\}$ and $\{6,9,11\}$
$56$ as a function of $\{3,10\}$ and $\{6,8,11\}$
$3$ as a function of $\{3,9\}$ and $\{2,5,7\}$
$11$ as a function of $\{1,5\}$ and $\{6,8,9\}$
$???$ as a function of $\{2,8\}$ and $\{1,3,4\}$

In every case, the number is

a multiple of either one of the first pair (same colour circle) or their sum or their difference.

So our final answer should be

even, at least.

But I also suspect that

modular arithmetic

might be involved, this being a clock and all.

  • 1
    $\begingroup$ Great start! I'm honored to have the great puzzle solver take a crack! $\endgroup$ Commented May 3, 2020 at 16:55
  • 1
    $\begingroup$ Added another hint if you'd like to try again $\endgroup$ Commented May 4, 2020 at 18:43

The answer should be:

A in Red(i.e. A should be written with the Red color)


Each number is coded with the base of its clock position. That is, 11 at position 1 is coded as 11 in base 1, 100 means the Binary equivalent of 100 and so on.
After we deduce each value of the number in the decimal number system, we see that there are 3 numbers each inside Red, Blue, Green and Yellow circles. Each number is also written in Red or Blue or Green or Yellow. The number written in Yellow is the product of the number written in Red and Blue.

Table of All numbers and their meaning:

The First column is the number in decimal system, the second column is the base in which it was originally written, the third column is the color of the circle around it and the fourth column is the color in which the number is written.

 2    - Base 1-   Green           Red
4 - Base 2- Yellow Blue
17 - Base 3- Red Red
7 - Base 4- Blue Red
6 - Base 5- Green Blue
91 - Base 6- Blue Yellow
13 - Base 7- Blue Blue
40 - Base 8- Yellow Yellow
51 - Base 9- Red Yellow
3 - Base 10- Red Blue
12 - Base 11- Green Yellow

From the table,

We can see that the final answer has to be in a Yellow circle. And as we already know that anything written in Yellow inside any colored circle is the product, we already know that our product will be 40. We know one of the numbers, in blue is 4 in decimal number system. So, the other number has to be 10 in decimal number system and has to be written in Red.
Converting 10 from decimal to base-12 will give A and that is the final answer.

  • 3
    $\begingroup$ 100 is 4 in binary $\endgroup$
    – JMP
    Commented May 4, 2020 at 19:56
  • 1
    $\begingroup$ oh lol. Yeah. Just a slight change in the answer then $\endgroup$
    – Sid
    Commented May 4, 2020 at 20:22
  • $\begingroup$ Why not just 'A'? $\endgroup$
    – JMP
    Commented May 5, 2020 at 4:44
  • 1
    $\begingroup$ Nicely done. I tried to do something with bases, but the base-1 deterred me because normally we don't use base 1 as it doesn't really make sense. I should have persevered and written down the list of numbers in the appropriate bases to spot connections. $\endgroup$ Commented May 5, 2020 at 8:24
  • 1
    $\begingroup$ Argh base 1 is weird. I thought it can only use the digit 0 and this really threw me off. Otherwise I would have done the exact table that you did. Well done! $\endgroup$ Commented May 9, 2020 at 1:26

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