Brain Teaser #84: Ancient Laurasian Formula & Numerical Sequences

[davilla]

It's time to study ancient Laurasian civilization. Here's an important formula that may or may not look familiar:

L = J'6oAL EL5J YCAQ JC5C 68oL -> 5'TYoA QELL |6T5 LLC5 J5dd i

Some common numerical sequences are encoded below:

J, d, E, o, Q, LL, Ld, L8, dQ, E2, 56, o8, J|2, JE8, T88, Y26, YA2, P2L, U2Q

L, h, 5, LY, oY, hQ|, Ld6|, 2QY|, EY2Y|, do EBV, h5L LEV, L9Y9 B9V, L odhY 99V

L, L, h, d, [, Y, Q, L[, hh, do, [2, 2/, e2, Lo2, hJh, dQ6, JdQ, SLY, L/[[, LSJQ

/, b, J, S, B, /Z, /9, h7, hQ, bo, 7h, 7e, Z6, J2, oY, YY, 22, S6, 6e, Qh, eo, BQ

/, 7, 2, //, /2, h7, 6/, 7/, Z/, J7, oY, 2/, S2, 97, e/, /V, /h/, /77, /J2, /2/

But something was lost in the transliteration. To make this scheme consistent, what diacritical mark should be added to the o?

Hint: The arrow is exponentiation.

 

[padraig]

dear god!

 

[marcus]

dear god!

come on padraig it is not that bad
the imaginary number i is carried over unchanged
and the first line says 1 = e raised to power 2 pi i
e is 2.71828... worked out in base 16 I suppose
pi is 6.283185... again I would guess in base 16 because

they are using base 16
and L = 1
h = 2
d = 3
[ = 5


I rather suspect that the third sequence is the Fibonacci numbers
1,1,2,3,5,8,13, 21,...

so that would make Y = 8
and Q = 13
and L[ would be 1 in the 16s place and 5 in the ones place, or 21
so it works out

 

[marcus]

It's time to study ancient Laurasian civilization. Here's an important formula that may or may not look familiar:

L = J'6oAL EL5J YCAQ JC5C 68oL -> 5'TYoA QELL |6T5 LLC5 J5dd i

Some common numerical sequences are encoded below:
...

L, L, h, d, [, Y, Q, L[, hh, do, [2, 2/, e2, Lo2, hJh, dQ6, JdQ, SLY, L/[[, LSJQ

...

I have used the edit to correct a mistake that davilla mentioned.
the euler formula 1 = e^2πi
and the fibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,

putting that in 16-base (hexadecimal) gives

1, 1, 2, 3, 5, 8, "13", 15, 22, 37, 59, 90, "14"9, 179, 262

this shows that the slash / must be the zero in the notation

/ = 0
L = 1
h = 2
d = 3

[ = 5
J = 6
o = 7
Y = 8
2 = 9

Q = 13
e = 14

 

[marcus]

L = J'6oAL EL5J YCAQ JC5C 68oL -> 5'TYoA QELL |6T5 LLC5 J5dd i

L, L, h, d, [, Y, Q, L[, hh, do, [2, 2/, e2, Lo2, hJh, dQ6, JdQ, SLY, L/[[, LSJQ

the euler formula 1 = e^2πi
and the fibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,

putting that in 16-base (hexadecimal) gives

1, 1, 2, 3, 5, 8, D = "13", 15, 22, 37, 59, 90, "14"9, 179, 262, 3 "13" "11", 63 "13", "10" 18, 1055, 1 "10" 6 "13"

this shows that the slash / must be the zero in the notation

/ = 0
L = 1
h = 2
d = 3
...
[ = 5
J = 6
o = 7
Y = 8
2 = 9
S = 10
6 = 11
Q = 13
e = 14

 

[marcus]

there may be inconsistent stuff mixed in
but that is a start on it
at least one sequence is the fibonacci
in base-16

somebody else see if they can take it further

 

[marcus]

L = J'6oAL EL5J YCAQ JC5C 68oL -> 5'TYoA QELL |6T5 LLC5 J5dd i

L, L, h, d, [, Y, Q, L[, hh, do, [2, 2/, e2, Lo2, hJh, dQ6, JdQ, SLY, L/[[, LSJQ

the euler formula 1 = e^2πi
and the fibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,

putting that in 16-base (hexadecimal) gives

1, 1, 2, 3, 5, 8, D = "13", 15, 22, 37, 59, 90, "14"9, 179, 262, 3 "13" "11", 63 "13", "10" 18, 1055, 1 "10" 6 "13"


/ = 0
L = 1
h = 2
d = 3
7 = 4
[ = 5
J = 6
o = 7
Y = 8
2 = 9
S = 10
6 = 11
9 = 12
Q = 13
e = 14
B = 15

the last sequence is the sequence of squares, 1, 4, 9, 16, 25, 36,...
but written in hexadecimal and with some alternative numerals
the three, which was d, can be turned front to back and appear as 6
the four can also be written 7

here is the sequence of squares:
L, 7, 2, L/, L2, h7, dL, 7/, [L, J7, o2, 2/, S2, 97, eL, L//, LhL, L77, LJ2, L2/

here is how this transliterates to hex
1, 4, 9, 10, 19, 24, 31, 40, 51, 64, 79, 90, "ten"9, "twelve"4, "14"1, 100, 121...

here is how this converts to decimal

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, ...

 

[marcus]

Translations of three of the 5 sequences

L, L, h, d, [, Y, Q, L[, hh, do, [2, 2/, e2, Lo2, hJh, dQ6, JdQ, SLY, L/[[, LSJQ

Fibonacci: 1, 1, 2, 3, ...

L, b, J, S, B, LZ, L9, h7, hQ, bo, 7h, 7e, Z6, J2, oY, YY, 22, S6, 6e, Qh, eo, BQ

bowling pin numbers, triangular numbers:
1, 3, 6, 10, 15, 21,...

L, 7, 2, L /, L2, h7, 6L, 7/, ZL, J7, oY, 2/, S2, 97, eL, L//, LhL, L77, LJ2, L2/

the squares:
1, 4, 9, 16, 25,...

 

[davilla]

Wow, lots of progress. There is one error in your table, but you are right on everything else. So, can anyone solve the puzzle?

To make this scheme consistent, what diacritical mark should be added to the o?

Incidentally, the Mayans used a base 20 number system with symbols using bars and dots. The dot is one and the bar is five, so the largest digit was written with four dots on top of three bars. Does anyone know of other civilizations using a base other than 10? Even the Romans grouped by 10.

 

[tribdog]

I remember when brain teasers were written in english.

 

[hemmul]

I remember when brain teasers were written in english.

 

[paul11273]

English has my vote.

 

[toosm:)ey]

Brain teasers?!? These are just cruel!

 

[davilla]

Sorry, I didn't intend any permanent damage to your eyes. It looks really complicated only because there's more information provided than is needed. The values used for e and pi were precise to 24 decimals, but that formula can be completely ignored if desired.

With a working interpretation, the toughest number chunching has already been done, although the work has gone unverified. That doesn't mean the answer is immediate, but it does require some human thinking in contrast to the more boring stuff a supercomputer can do.

If catching the lingering mistake isn't enough to solve the problem, there are extra clues in symbols that have not been identified. Marcus even mentioned that the d can be flipped front to back, a fact that he omitted from the table.

 

[davilla]

Second Hint:

Confused by apparent contradictions in their decoding scheme, anthropologists went back to the site to discover that the J in the most recent carvings, including the formula, was written with one stroke, whereas the J that they had determined to represent six was written with two.

 

[marcus]

hi davilla, that's a fascinating puzzle and I am surprised that no one else has stepped in and furthered the progress already made on it

basically I would like to share the fun
and not be the only person responding

when other people don't try it suggest to me that they arent interested
so I don't want to just talk to myself

if other people chip in and help then I will get back to it too
because I can see that I have a mistake in either 4 or 7 and
that there are more things to discover about the way they have
of writing numerals

like zero is /
and one is / with an underline, which looks like L or else like /

so maybe this is a Laurasian rule that the next number is made by putting
an underline

stuff like that

and I have a guess as to which sequence involves prime numbers

but if no one else tries to answer then I should not just be talking to a blank wall

anyway, brava and kudos for the puzzlemaking

 

[PRodQuanta]

I have no idea. I've been pondering. My guess is an anunaasika, but that's purely a guess.

The only thing I can think of is that the e is a small c with a diacritic bar through it.

Hey, I've gave it all I got, to no avail. Good luck.

Paden Roder

 

[gnome]

Hmm... sorry I didn't even notice this one until now. And I can't work on it now. Davilla mentioned on the first page something about other number systems, but it's not clear whether that was a hint or just a point of general interest. So I'll just remind everyone that the Babylonians and Sumerians used a sexagesimal (base 60) number system in case it might be relevant to solving the rest of this puzzle.

 

[marcus]

It's time to study ancient Laurasian civilization. Here's an important formula that may or may not look familiar:

L = J'6oAL EL5J YCAQ JC5C 68oL -> 5'TYoA QELL |6T5 LLC5 J5dd i

Some common numerical sequences are encoded below:

J, d, E, o, Q, LL, Ld, L8, dQ, E2, 56, o8, J|2, JE8, T88, Y26, YA2, P2L, U2Q

L, h, 5, LY, oY, hQ|, Ld6|, 2QY|, EY2Y|, do EBV, h5L LEV, L9Y9 B9V, L odhY 99V

L, L, h, d, [, Y, Q, L[, hh, do, [2, 2/, e2, Lo2, hJh, dQ6, JdQ, SLY, L/[[, LSJQ

/, b, J, S, B, /Z, /9, h7, hQ, bo, 7h, 7e, Z6, J2, oY, YY, 22, S6, 6e, Qh, eo, BQ

/, 7, 2, //, /2, h7, 6/, 7/, Z/, J7, oY, 2/, S2, 97, e/, /V, /h/, /77, /J2, /2/

But something was lost in the transliteration. To make this scheme consistent, what diacritical mark should be added to the o?

Hint: The arrow is exponentiation.

I think the problem is solved---at least in this sense: we know that the last 3 sequences are Fibs, triangular numbers, and squares.
We know the system is hexadecimal with these symbols:


/ = 0
L = 1
h = 2
d = 3
7 = 4
[ = 5
J = 6
o = 7
Y = 8
2 = 9
S = 10
6 = 11
9 = 12
Q = 13
e = 14
B = 15

That constitutes a complete decoding, as far as I can see. A mistake mentioned by davilla has been corrected.

However there is still the formula to check, plus these two sequences

J, d, E, o, Q, LL, Ld, L8, dQ, E2, 56, o8, J|2, JE8, T88, Y26, YA2, P2L, U2Q

L, h, 5, LY, oY, hQ|, Ld6|, 2QY|, EY2Y|, do EBV, h5L LEV, L9Y9 B9V, L odhY 99V

The J in the first sequence may just be a typo. It should probably be h (to stand for 2, the first prime number). My concern about this is that
if the rest turn out to be consistent then nothing new is learned
and if they turn out to be inconsistent then its a mess. because we already have a complete decoding.

Like suppose these Laurasians thought that they were writing the prime numbers, but they left out some? then the first sequence is going to screw us up!

in any culture, the primes are:
2,3,5,7,11,13,17,19,23, 29, 31, 37, ...

when I look at the first Laurasian sequence I don't see "11" which they write 6.

h, d, [, o, (6, for 11, is missing) Q, LL, Ld, (Lo, for 23, missing) (LQ, for 29, missing) LB,...

so there are a lot of primes in their sequence but not all----there are gaps

 

[gnome]

Some additions to marcus' decoding:

/ and | = 0
L and /_ = 1
h and J = 2
d, T, b and | = 3
7, 6 and T = 4
[, Z and E = 5
J and 5 = 6
o and 8 = 7
Y = 8
2 = 9
S = 10
6 = 11
9 = 12
Q = 13
A and e = 14
B = 15

Of course, some of these apparent dual meanings may simply be typos or arithmetic errors on the part of the Laurasians.

As for the primes, I guess they left out some figuring we didn't need all of them. On the other hand, they included o8=77h=119d which doesn't belong there (unless 8 sometimes means 1 or 15).

The second sequence is factorials, except they listed LY=18h=26d instead of 24d (unless Y sometimes means 6).

The J is not a typo, that's the dual meaning that davilla mentioned (turns out to be just one of many dual meanings). In the formula, and the prime sequence J has to be 2; in the fibonacci and prime sequences, it is 6.

So now I think everything is decoded except the underlying basis for the code scheme itself, which, if someone finds it, will presumably lead to the solution.

 

[davilla]

This must be much more difficult than I thought. (Edited:) gnome, there are several errors in your most recent table, two of which you're aware of but haven't identified as mistakes on your part.

some of these apparent dual meanings may simply be typos or arithmetic errors on the part of the Laurasians

Thanks for the humor, but so far I haven't seen any errors in the original problem. The dual J was supposed to be tricky, and perhaps too much so. It is not a true conflation because the two were written differently.

As for the primes, I guess they left out some figuring we didn't need all of them.

The savageness of this human (?) culture is clearly represented in their disregard for satanic numbers! Or perhaps there is some temporal significance to the drastic difference between the simpler sequences and the more recent. Try to determine the pattern based on the symbols you know to be correct.

LY=18h=26d instead of 24d

LY = 16 + 8 = 24

So now I think everything is decoded except the underlying basis for the code scheme itself, which, if someone finds it, will presumably lead to the solution.

Correct.

 

[gnome]

(me):...they listed LY=18h=26d instead of 24d (unless Y sometimes means 6).

I guess I was more tired than I realized last night.


How are you joining the underline to the slash in that alternate symbol for 1: "/_"?

(davilla)There are a couple of errors in the table, both of which you're aware of but haven't identified as mistakes on your part.(my italics)

I don't understand what you're saying here. Mistakes in my table or yours?

Are these the errors to which you are referring:
In the square sequence 6"/_" = B1h or 177d. Presumably it should have been 7"/_" = 41h = 49d.

In the formula, 5'TYoA ... should be 5'7YoA ...

 

[gnome]

OK, before I waste too many hours trying to find some elaborate scheme for translating ascii codes:

davilla, suppose we simply add a ° to the o to produce an 8?

 

[davilla]

some elaborate scheme for translating ascii codes

Laurasia was the northern landmass of North America and Eurasia that formed from the splitting of Pangaea, existing two to one hundred million years before the first humans walked the Earth. A descendant of Bell Labs' Teletype Wide-area Exchange, ASCII was first published as a standard about 5 million years thereafter. My first brain teaser may have been cruel, for which I apologize, but I assure you that it is not whimsically reasoned or arbitrarily justified.

Thanks for the Babylonian and Sumerian reference. I'll have to read up on it.


Third Hint:

There are only two characters that appear to have double meaning. One was named in the second hint, and the other was discovered early and explained by marcus.

 

[gnome]

OK, I see my errors. The Laurasian 8 is actually "15", not 7, the | is unambiguously 0 (never 3 as I originally thought), and the Laurasian T unambiguously represents 4. In addition, we can add P and U (both of which represent 12) to the codification.

Therefore the primes listed (in decimal numbers) are:
2 3 5 7 13 17 19 31 61 89 107 127 521 607 1279 2203 2281 3217 3229

And with T representing 4, now 5'TYoA ... correctly represents 2Π.

So now we have:
/ and | = 0
L and /_ = 1
h and J = 2
d and b = 3
7 and T = 4 ...edited to remove the 6 that mysteriously slipped in there.
[, Z and E = 5
J and 5 = 6 ...edited font
o = 7
Y = 8
2 = 9
S = 10
6 = 11
9, P and U = 12
Q = 13
A and e = 14
B and 8 = 15

Why remains to be seen.

(This is fascinating.)

 

[gnome]

Got it.

You must add something that looks like T or 7 to the o.

Thank you davilla. That was superb.

PS: how did you make that alternate symbol for 1 ("/_")? I still can't find a code to produce that.

 

[davilla]

You must add something that looks like T or 7 to the o.

Correct! I was thinking of just a bar, but I guess it makes sense to connect it to the o somehow.


Tack-on Teasers:

For an extra point, name the sequence generated employing, in part, a mapping of h -> to the first sequence in the problem.

For an extra point, express U -> Z'Y in two characters.

 

[gnome]

Does "mapping of h -> to the first sequence ..." mean
h -> J
h -> d
h -> E
and so on ?

 

[davilla]

Does "mapping of h -> to the first sequence..." mean
h -> J
h -> d
h -> E
and so on?

Yes, or more formally, a mapping of the function lambda(x) = h -> x to the first sequence.

 

[gnome]

Do the "prime powers of 2" have some particular significance, or did you have something else in mind?

 

[davilla]

Do the "prime powers of 2" have some particular significance?

As noted, there are primes missing, so they do not yield all of the prime powers of 2.

 

[gnome]

As noted, there are primes missing, so they do not yield all of the prime powers of 2.

Even the sequence of all integers employs "in part" the numbers included in this mapping. Your wording describes a sequence that includes numbers in addition to those generated by the mapping. Did you mean it to be interpreted differently?

 

[davilla]

Name the sequence generated employing, in part, a mapping of h -> to the first sequence in the problem.

What I mean is, find the significance of the first sequence. To that end, I have provided a formula that will prove useful, without giving too much away.