This puzzle is based off the What is a Word™ and What is a Phrase™ series started by JLee and the likewise inspired What is a Number™ series.

If a number conforms to a certain rule, I call it an ES Number™. Otherwise, I call it an OS Number™ Use the following examples to find the rule:

ES Numbers™ and OS Numbers™

Here is a CSV:

ES Numbers™,OS Numbers™

The puzzle relies on the series' inbuilt assumption, that each number can be tested for whether it is an ES Number™ or an OS Number™ on its own. In particular, a number's relationship to other numbers in the sequence is irrelevant.

These are not the only examples of ES or OS Numbers™, more can be found.


I made a mistake in my ES Numbers™ list. 24 is actually an OS Number™. In order to make something similar to the original, I have included 96 which is an ES Number™. Unfortunately this makes the puzzle less cruel. ;)

Once again, sorry for the time you may or may not have wasted. I double-checked the puzzle and still didn't pick up the mistake. Now it is solvable in the intended way. (I've triple checked that)

  • 1
    $\begingroup$ E and O most likely stand for even and odd $\endgroup$ – Beastly Gerbil Oct 29 '16 at 9:26
  • 1
    $\begingroup$ There is an issue in your numbers, the last ES number in the table is 8406269402, but in the CSV it's 8406268402. $\endgroup$ – Alenanno Oct 29 '16 at 10:01
  • $\begingroup$ Does E'S stand for anything $\endgroup$ – user17008 Oct 30 '16 at 12:21
  • $\begingroup$ @ev3commander: Veryl likely, but that's for us to find out. $\endgroup$ – M Oehm Oct 30 '16 at 13:25

An ES Number™ is a number that:

Uses an even number of segments when shown in a seven-segment display.

ES and OS stand for

Even-segmented and odd-segmented

| improve this answer | |
  • $\begingroup$ @boboquack there are three 2s, so the count is odd (ones and zeros are out because 1^2=1 and 10^2=100). If this isn't correct, I think we need more examples. $\endgroup$ – ffao Oct 31 '16 at 23:07
  • $\begingroup$ It is correct, but that wasn't my reasoning. See whether you can find a simpler way to express it and I may accept your answer. Problem is, Rubio posted an answer that says the same thing so I can't accept your answer. There is a simpler way... $\endgroup$ – boboquack Nov 1 '16 at 2:00
  • $\begingroup$ @boboquack I looked at Rubio's answer and saw that my earlier rule failed for 1234567. This one shouldn't... $\endgroup$ – ffao Nov 1 '16 at 3:27
  • $\begingroup$ Correct - I've added a bit more info in an edit. $\endgroup$ – boboquack Nov 1 '16 at 3:39
  • $\begingroup$ That'll teach me to step away for a few hours. :) Congrats ffao for figuring out what the pattern means! $\endgroup$ – Rubio Nov 1 '16 at 6:24

I wonder whether ES numbers are simply those

whose digits have an even sum (hence ES).

There is a single obstacle to this, which is that

the digits of 48 add up to 12, which is even.

Is it possible that that one is a mistake?

Perhaps it was intended to be 12 (half the number in the ES column) rather than 48 (twice the number in the ES column).

| improve this answer | |
  • $\begingroup$ No, that 48 was intentional. It was purposely there to stop such a solution. (It was probably a bit cruel of me to create the numbers like that). +1 anyway for spotting the problem. $\endgroup$ – boboquack Oct 29 '16 at 21:27
  • $\begingroup$ Well, this is why I didn't post an answer. I discovered it when checking all the sums to test that, but... yeah. Very mean, @boboquack! (No, kidding, nicely done.) $\endgroup$ – Alenanno Oct 29 '16 at 22:21
  • $\begingroup$ @Alenanno I think digital sums are too easy for the types of puzzlers that populate this website. $\endgroup$ – boboquack Oct 29 '16 at 23:33
  • $\begingroup$ Me too, which is why I was a little surprised to find things looking so much as if they were the answer :-). $\endgroup$ – Gareth McCaughan Oct 30 '16 at 0:19
  • $\begingroup$ See my edit: it doesn't change the 48 but does change the 24. $\endgroup$ – boboquack Oct 31 '16 at 9:54

Here's a solution, though if it's the intended one I'll be surprised.
Nevertheless, since it works to distinguish between the given ES and OS lists, I'll post it.

First let's assign ...

a PSEValue of 0 to the digits 1,4,6,9,0.
a PSEValue of 1 to the digits 2,3,5,7,8.

I don't have a particular justification for these PSEValues, other than the fact that these choices happen to make the pattern work, which is why I assume this isn't the actual solution.

Then we use that to find, for each number in the lists,

a sum of the PSEValues assigned to each digit in the number.

This gives us:

$$\begin{array}{|c|c|}\hline\Bbb{ES\ Numbers}^\text{TM}&\Bbb{OS\ Numbers}^\text{TM}\\\hline^{4: 0 → 0}&^{5: 1 → 1}\\^{96: 0+0 → 0}&^{48: 0+1 → 1}\\^{343: 1+0+1 → 2}&^{434: 0+1+0 → 1}\\^{2048: 1+0+0+1 → 2}&^{1024: 0+0+1+0 → 1}\\^{15679: 0+1+0+1+0 → 2}&^{14679: 0+0+0+1+0 → 1}\\^{86420: 1+0+0+1+0 → 2}&^{97531: 0+1+1+1+0 → 3}\\^{1234567: 0+1+1+0+1+0+1 → 4}&^{6543210: 0+1+0+1+1+0+0 → 3}\\^{543543543: 1+0+1+1+0+1+1+0+1 → 6}&^{210210210: 1+0+0+1+0+0+1+0+0 → 3}\\^{3597175319: 1+1+0+1+0+1+1+1+0+0 → 6}\ &^{1354279860: 0+1+1+0+1+1+0+1+0+0 → 5}\ \\^{8406268402: 1+0+0+0+1+0+1+0+0+1 → 4}\ &^{9068731254: 0+0+0+1+1+1+0+1+1+0 → 5}\ \\\hline\end{array}$$

So the numbers are called ES Numbers™ or OS Numbers™ depending on

whether the sum of the PSEValues for their digits is an Even Sum or an Odd Sum.

upgraded MagicValues to PSEValues, to fit the revised puzzle.

| improve this answer | |
  • $\begingroup$ It's magic... not quite. There is a more logical solution. Anyway, nice TEX table. If it was about magic values the problem would be EMVS and OMVS numbers ;) $\endgroup$ – boboquack Oct 31 '16 at 9:04
  • $\begingroup$ I had to try. And, well, it does fit .... ;) (And, incidentally, is the only set of MagicValues for the digits that makes the ES/OS lists work. Fun!) $\endgroup$ – Rubio Oct 31 '16 at 9:08
  • $\begingroup$ Did you check with a computer program? $\endgroup$ – boboquack Oct 31 '16 at 9:22
  • $\begingroup$ Yes I did indeed. $\endgroup$ – Rubio Oct 31 '16 at 9:23
  • $\begingroup$ See my edit. Sorry for making you go to the trouble of writing a computer program for nothing. $\endgroup$ – boboquack Oct 31 '16 at 9:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.