# What is an ES Number™?

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:

Here is a CSV:

ES Numbers™,OS Numbers™
4,5
96,48
343,434
2048,1024
15679,14679
86420,97531
1234567,6543210
543543543,210210210
3597175319,1354279860
8406268402,9068731254


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.

EDIT: SORRY TO EVERYONE WHO ATTEMPTED THIS PUZZLE

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)

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

• @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. – ffao Oct 31 '16 at 23:07
• 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... – boboquack Nov 1 '16 at 2:00
• @boboquack I looked at Rubio's answer and saw that my earlier rule failed for 1234567. This one shouldn't... – ffao Nov 1 '16 at 3:27
• Correct - I've added a bit more info in an edit. – boboquack Nov 1 '16 at 3:39
• That'll teach me to step away for a few hours. :) Congrats ffao for figuring out what the pattern means! – 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).

• 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. – boboquack Oct 29 '16 at 21:27
• 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.) – Alenanno Oct 29 '16 at 22:21
• @Alenanno I think digital sums are too easy for the types of puzzlers that populate this website. – boboquack Oct 29 '16 at 23:33
• Me too, which is why I was a little surprised to find things looking so much as if they were the answer :-). – Gareth McCaughan Oct 30 '16 at 0:19
• See my edit: it doesn't change the 48 but does change the 24. – 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.

• 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 ;) – boboquack Oct 31 '16 at 9:04
• 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!) – Rubio Oct 31 '16 at 9:08
• Did you check with a computer program? – boboquack Oct 31 '16 at 9:22
• Yes I did indeed. – Rubio Oct 31 '16 at 9:23
• See my edit. Sorry for making you go to the trouble of writing a computer program for nothing. – boboquack Oct 31 '16 at 9:54