Step 9 Work in progress

Clue updates
The bottom number was replace from 7 to 3 and the factors adjusted accordingly.

Thoughts: He avoided using 5 because it's used on the right part and someone new to the game would link the 2 together. 6 and 4 were skipped because they're not prime so it was either 3 or 2.

The change was done because we were spending time looking in the wrong direction, it seems that average of 4 and 6 = 5 and average of 7 and 9 = 8 might not be the place to look at.

Considering that, since the arrows didn't change, what do they mean? They could potentially just be an alien equivalent of thumbs up (3, 7) and thumbs down (9) indicating one is prime and good and the other is not. The seems somehow redundant with the bottom part where T=8.

The arrows were removed, we no longer have to figure them out.

Cerbere
What's the theory behind -123103, -329939?

LightningWBToday at 1:17 PM
From the top I got 7 because of factor count plus one and then (7^3)^3 then took the first 6 digits so it is in our playable area. I then did the same with 5 and took the first 6 digits.

DeffinataleeToday at 1:36 PM
especially in the new puzzle, we see powers of three

I think this is what toro wants us to look at

different powers of primes

Clue v2
Since we got an updated version of the clue, some of the previous ideas don't apply anymore.

Definitions
T: number of factors of a number. Since the factors can be positive or negative, T should be an even number. Why are there negative factors? AFAIK, factoring has always been done with positive numbers.

U: what is U? It seems to be the result of operations done in the big box and it seems to be a prime number, or at least it should have 8 factors.

Theory 1
125 refers to either the first 125 numbers of a sequence or the first 125 digits of that sequence similar to the Fibonacci clue. That sequence might be the prime numbers.

What do we do with all these numbers to get coordinates?

1.1
The 125th prime number is 691. 691^3 = 329939371. maybe we could split that in 2 parts but that seems to be pretty random.

3,29939371

32,9939371

329.939371

3299,39371

32993,9371

329939,371

3299393,71

32993937,1

691^3 in base 9 758748651 split up

7 58748651

75 8748651

758 748651

7587 48651

75874 8651

758748 651

7587486 51

75874865 1

converted back to base 10

7 28612324

68 4697479

620 445951

5587 293590

... I didn't convert the rest, I think y becomes too big so the last 2 are possible candidates

1.2
The first 125 first prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691

1.3
The first 125 digits from prime:

23571113171923293137414347535961677173798389971011031071091131271311371391491511571631671731791811911931971992112232272292332

Theory 2
We know the world boundaries are at 500000 positive and negative. The circle in the clue is located near the middle line and close to the bottom of the map. What if we need to find the biggest prime^3 that fits in the world?

int(500000^(1/3)) = 79

By luck, 79 is prime.

Coordinates might be (-79 -493039)

As a coincidence, 7 and 9 are the 2 numbers in the left part of the upper clue box. Left part = x, Right part = y?

The up and down arrows might indicate to swap the numbers positions so instead of having 97 which would be outside the box, we have 79.

Status: location in an ocean

Theory 3
What if the arrows mean to do an operation on the base number. For example, 7 + T(7) = 11 and 9 - T(9) = 3. Both numbers are prime.

11^3 = 1331

factors 1, 11, 121, 1331

3^3 = 27 factors

1, 3, 9, 27

Then what?

There's a big flaw with this, we already have an operator for adding so why use arrows instead?

Theory 4
Phineas the Ferb

My theory was that the x coordinates could be given from U^4/125 because I thought the ratio between U and 125 had significance U/(5*5*5) = x coords/U^3 U/125 * U^3 = x x = U^4/125(edited)

I tried all the prime numbers I could think of up to 125, but none came out as integers.

0.128, 16

0.648, 27

5, 125

19.208,  343

117.128, 1331,

228.488, 2197

668.168, 4913

1042.568, 6859

2238.728, 12167

5658.248, 24389

7388.168, 29791

14993.288, 50653

22606.088 ,68921

27359.408,79507

39037.448,103823

63123.848,148877

96938.888,205379

110766.728，226981

161208.968，300763

203293.448，357911

227185.928，389017

311600.648，493039

379666.568，571787

501937.928，704969

708234.248，912673

(from here on, the y coordinates are too large)

832483.208，1030301

900407.048，1092727

1048636.808，1225043

1129265.288，1295029

1304378.888，1442897

Theory 5
Something I haven't explored is T(x) = 5 (average of T(7) and T(9). Here's some thoughts but I don't know what to do with this.

My understanding is that for every positive factor we have it's negative counterpart so T has to be even.

T(x) = 5 is therefore not possible.

Assuming we ignore negative factors, where would that get us?

To avoid confusion, let's define S as the number of positive factors.

Is there a type of number such that we can get S(x) = 5?

Let's look at what we already know.

x is a prime number

S(x^3) = 4

1 * x^3

x * x^2

What about x^2

S(x^2) = 3

1 * x^2

x * x

S(x^4) = 5

1   * x^4

x   * x^3

x^2 * x^2

So it appears at first glance that S(x^n) = n+1 ; for n>=2

psi in this clue is the same as psi in the previous clue and we need to use the location formula to get the coordinates
n in this clue is different than n in previous clue.

nprevious = in game year

ncurrent = number of factors of a number

psi3 has 8 factors so psi has to be prime.

The quadrants at the bottom of the clue indicate that the coordinates we find will be in the bottom left part of the map.

theory 1
> 7 && < 9

could it be indicating 8?

1,2,4,8,-1,-2,-4,-8

n=8

theory 2
n > 4 && n < 6

n = 5

With the given examples, n = 5 is not possible since we use the negative numbers n is always even.

Unlikely that this is the link between the left box and the right box with 53. Probably just a coincidence.

theory 1
see theory 2 above.

theory 2
this is just an example of a valid psi, 5 is prime and shown as cubed.

Theory 1
Since the main box has 5*5*5 pointing out to Ψ and we have Ψ*Ψ*Ψ as a prime number, it's possible that Ψ = 5 is the solution.

If we apply the sign change for the bottom left we get:

(-199995,-100005)

It's also possible that as in the previous clue this has to be checked on day 33.

(-199995,-100005) day 33

Status: Nothing but not tested on day 33.

Theory 2
We're looking for Ψ that is a prime number. Since in the previous clue Ψ is derived from ᑎ (game year), the we might be looking for a game year where the resulting Ψ is a prime number.

The next few prime Ψ would be on there game years.

133    104991    195009    1064095009

141    106289    243711    1160393711

223    186417    363583    1073713583

297    165247    434753    890734753

Apply negative signs and possibly check on day 33 as in previous clue.

Status: Nothing at coordinates of year 133 and day 33

Theory 3
Not really a theory, just a though. Could it be possible that the Fibonacci sequence is involved? If so, why and how would it relate to this puzzle?

Theory 4
psi = 2 because 8 is between 7 and 9. 8 = 2^3 and 2 is prime.

Status: untested

Theory 5
aleKaetor

maybe we take the current year and find its number of unique factors

Example:

Current year is 140, so we decompose that into factors, I think that gives 12 different ones so that would be 24 if we take the negative numbers into account, we plug that into the equation and we see what happens.

One of the problems with this approach is that there's already a very limited number of years when psi is prime, this reduces the option a lot so we'd probably have to wait a very long time before we'd have a shot at checking if the solution is valid and if we miss it, we're done for a long time. From a puzzle design perspective I don't think that's a good idea.

Status: untested.