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In the world of Metsyssisab, there were three kingdoms, ruled by King Owt, King Eerht, and King Ruof.

One day, a fourth, evil king, King Evif, who ruled a small island, looked to control all of the kingdoms. He challenged the three kings to a mathematical challenge. The three kings agreed. If they won, King Evif would flee Metsyssisab. If they lost, King Evif would take control of all their kingdoms. The rules of the competition as stated by King Evif were:

  1. Each of the three kings were to be given a digital display with 13 parts.
  2. Each part of the digital display could show as a single-digit number. Each segment could be lit up or turned off. The segments could ONLY show a number, and not anything else.
  3. The three kings were each to arrange their own individual display so that by lighting up any number of consecutive segments, they could count up to 13. (For example, a solution with too many segments is 10121345611789, because it's possible to light up the numbers 1 through 13 by using connected segments.)
  4. There was one catch. The sum of all the numbers shown on all of the segments had to be less than 20.
  5. The kings could not signify numbers by using anything else except the numbers displayed on the segments, and the actual numbers must be displayed, not the sum of or the number of numbers displayed.

The three kings each figured out how to arrange their display to satisfy King Evif's demands, and he left Metsyssisab. Can you?

Minor Hint:

There is a bit of numerical trickery going on here. I strongly suggest you figure that out first, or you won’t get too far.

Minor Hint #2:

The kings will each have different answers, for a clear reason once you figure out what's going on. What will work for one king won't work for another.

Minor Hint #3:

Take a look at the kings' and the world's name.

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  • $\begingroup$ Just out of curiosity does the display show the numbers like a typeface font (e.g 1,2,3, etc.) or is it more like a digital clock display? $\endgroup$ – gabbo1092 Oct 29 '18 at 16:50
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    $\begingroup$ @gabbo1092 I was assuming a typeface font, but if you're using the difference as a way to solve the puzzle, it's not based on that. $\endgroup$ – Excited Raichu Oct 29 '18 at 16:51
  • $\begingroup$ Do you mean I have to give 3 different ways of arranging the parts? $\endgroup$ – Wais Kamal Oct 29 '18 at 17:04
  • $\begingroup$ @WaisKamal somewhat, but not exactly... $\endgroup$ – Excited Raichu Oct 29 '18 at 17:11
  • $\begingroup$ It was (to me) ... less than perfectly clear ... that we were required to find what the three kings did rather than to find our own solution. $\endgroup$ – Gareth McCaughan Oct 30 '18 at 14:01
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Once we

read the kings' and the land's names backwards, it is clear they are using different bases for the solution.

When King Evif

states that the numbers up to 13 shall be displayed, he means eight, as ${13}_5=8_{10}$.
Also, the limit on the sum of digits is ten, as ${20}_5={10}_{10}$.

Possible solutions that fulfill these requirements are:

King Otw:

11101000 for base 2
11101000 = 1
11101000 = 10
11101000 = 11
11101000 = 100
11101000 = 101
11101000 = 110
11101000 = 111
11101000 = 1000

King Eerht:

20112210 for base 3
20112210 = 1
20112210 = 2
20112210 = 10
20112210 = 11
20112210 = 12
20112210 = 20
20112210 = 21
20112210 = 22

King Ruof:

12011013 for base 4
12011013 = 1
12011013 = 2
12011013 = 3
12011013 = 10
12011013 = 11
12011013 = 12
12011013 = 13
12011013 = 20

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  • $\begingroup$ (rot13) Guvf vf ernyyl pybfr, naq lbh'ir tbg gur tvzzvpx qbja. Whfg znxr fher gb erzrzore jub vf punyyratvat gurz! $\endgroup$ – Excited Raichu Oct 29 '18 at 17:29
  • $\begingroup$ @ExcitedRaichu, abg fher jung lbh ner ersreevat gb, nf gurer vf n fbyhgvba sbe onfr svir nf jryy $\endgroup$ – elias Oct 29 '18 at 17:41
  • $\begingroup$ xvat rivs vf gur bar jub naabhaprq gur punyyratr, vapyhqvat gur ahzoref hfrq jvguva vg $\endgroup$ – Excited Raichu Oct 29 '18 at 17:42
  • $\begingroup$ That's it! (I'm just going to edit in 8 for the base 2.) $\endgroup$ – Excited Raichu Oct 29 '18 at 18:01
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    $\begingroup$ (not rot13, just actual gibberish) hyxf bjy f oikrtr ommijoy plefq hurcf pimlltert hxhflloik hung ibh erzizry nubmn! $\endgroup$ – Bass Oct 29 '18 at 19:02
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The description is (perhaps deliberately) less than perfectly clear. It seems to me that

it would be consistent with the description to count in unary. So each of the 13 segments just shows a "1" or even a mere vertical tally-stroke, and then we represent the number n by lighting up n of the segments.

[EDITED to add:]

In view of the significance that we have learned attaches to the kings' names, I am adopting for the duration of this answer the pseudonym King Eno.

Alternatively

we could work in Roman numerals. We need segments IXVIII for this, totalling 19.

[EDITED to add:] Oops, no, I think the second of these doesn't quite work because of

the requirement for the illuminated segments to be consecutive. We could, cheekily, replace the X by a V above an inverted V (I have heard it claimed that this is actually the origin of the X in Roman numerals). This also reduces the sum of the numbers to 14 :-).

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  • $\begingroup$ whoops, nice finding a little oversight on my part there. I'll add a clarification. $\endgroup$ – Excited Raichu Oct 29 '18 at 17:05
  • $\begingroup$ Even with your clarification, I claim that my first answer should be acceptable. The edit I've just made may make it more obvious why. $\endgroup$ – Gareth McCaughan Oct 29 '18 at 17:44
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Solution:

1 The name of the world Metsyssisab spells basis system backwards, and each of the kings name is a number backwards (Two, Three and Four). Binary is a system of base 2 numbers. The Ternary numeral system is base 3. There is also a number system called base 4. With these number systems each king can compose a solution to King Evif's challenge.

King Owt's:

using 1s and 0s to represent some of the numbers in binary.

1110001101001

Numbers 10 and 11 can be displayed by simply turning on an adjacent 1,0 or 1,1 respectively. All other numbers can be represented as binary, 1 is 1, 10 is 2. 11 is 3, etc. With the full display adding only to 7

Finishing King Eerht's and King Rouf's now. Will update soon.

King Eerht's:

King Rouf's:

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  • 1
    $\begingroup$ This is the closest answer so far. $\endgroup$ – Excited Raichu Oct 29 '18 at 17:05
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Probably not allowed but:

You could do it in two digits of $1$ and $8$ and put bits of tape (or other material) over the $8$ to hide segments and make the other numbers out of it.

Partial Alternative:

I think the numerical trickery may be to do with using non-symmetrical numbers upside down to create symbols which can represent the values 10, 11, 12, and 13 in base 13.

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0
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Is it that simple:

1111111111111

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  • $\begingroup$ If so, then I got there before you did :-). $\endgroup$ – Gareth McCaughan Oct 29 '18 at 16:59
  • $\begingroup$ Yeah, you beat me by abt 2 mins :) $\endgroup$ – Wais Kamal Oct 29 '18 at 17:01

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