1x1 Indiana Jones' mysterious wooden cube

[Tito_H]

I received a package from Indy. Inside was a letter and a USB. The letter read, "Dear Tito, Thank you for all your help the other day. I am currently in Ommeren, a small village in the Netherlands, and am in the middle of searching for treasure using a map that is said to have been left behind by a German soldier.

I went to the location marked on the map and while digging, I came across an old wooden cube. I believe this cube has something to do with the treasure, so I've been looking into it, but I haven't been able to solve the mystery yet. So, I'd like to ask you again, I've enclosed a USB recording of what the cube looks like, so could you please analyze it? Thank you very much."

I immediately looked at the image stored in the USB, and the cube looked like this:

- It was a 1x1x1 cube like the one below.



- The surface had corroded and faded over the years, but it was painted in 3 different colors, and X-ray analysis revealed that Greek letters, arrows, and numbers were engraved on each face, as shown below.

U face: α → 270
F face: α → 450
R face: β → 75
B face : β → 120
L face: γ → 12
D face: γ → 200

- I also saw the words "+3 turns" on the F face.

(U, F, R faces view)

(B, L, D faces view)

Based on the above information, I thought the problem was to guess in which direction this cube needs to be rotated three times (each 90 degrees) so that the initial digits (α, β, γ) each become the engraved digits, applying some math formula. However, since we don't know the initial 3 digits, the 3 rotation directions, or the calculation formula, it doesn't seem like it will be easy to solve. Let's think about it a bit more...

Can you solve it, readers?

 

[Tito_H]

I reasoned as follows:

1. I factorized the numbers engraved on the cube.

U face: α → 270 = 2x3x3x3x5
F face: α → 450 = 2x3x3x5x5
R face: β → 75 = 3x5x5
B face : β → 120 = 2x2x2x3x5
L face: γ → 12 = 2x2x3
D face: γ → 200 = 2x2x2x5x5

If this is arranged on the floor plan of the cube, it looks like this:
(U, L, F, R, B, D are the faces of the cube.)

(Floor plan)
(U)2x3x3x3x5
(L)2x2x3 (F)2x3x3x5x5 (R)3x5x5 (B)2x2x2x3x5
(D)2x2x2x5x5

From the results, the following inferences were drawn:

1-1. The engraved number is the product of the prime numbers 2, 3, and 5, and these prime numbers correspond to α, β, and γ.

1-2. The prime numbers 2 and 3 are on the L side, and the prime numbers 2 and 5 are on the D side, and both are γ, so γ=2.

1-3. From 1-2 above, α=3 and β=5, or α=5 and β=3.

2. The calculation formula is estimated to be:

new number = (original number) x (number of overlaps per 90-degree rotation of the cube)

3. Based on the above 2, the number of L face terms (3) was compared with the number of L face terms that change with cube rotation (p). As a result, it was found that of the 27 (3x3x3) rotation combinations, there are only 4 where p = 3:

x → y → x, x → z → x, y → z → x, and z → y → x.

Furthermore, since the cube is engraved with “+3 turns”, the reverse rotation (x', y', z') was excluded from consideration.

4. Based on the above, I verified by entering numbers into α, β, and γ.

(Floor plan)
Upper
Left Front Right Back
Down

<Case1: α=3, β=5, γ=2>

Pattern x→y→x:

3
2 3 5 5
2

↓ x rotation

3x3
2 2x3 5 3x5
2x5

↓ y rotation

3x3
2x2x3 2x3x5 3x5x5 2x3x5
2x5

↓ x rotation

2x3x3x3x5
2x2x3 2x2x3x5x5 3x5x5 2x3x3x3x5
2x2x3x5x5
This doesn't match the engraved number, so it's out.

Pattern x→z→x:

3
2 3 5 5
2

↓ x rotation

3x3
2 2x3 5 3x5
2x5

↓ z rotation

2x3x3
2x2x5 2x3 3x3x5 3x5
2x5x5
At this point, the L face does not match, so this is out.

Pattern y→z→x:

3
2 3 5 5
2

↓ y rotation

3
2x3 3x5 5x5 2x5
2

↓ z rotation

2x3x3
2x2x3 3x5 3x5x5 2x5
2x5x5

↓ x rotation

2x3x3x3x5
2x2x3 2x3x5x5x5 3x5x5 2x2x3x3x5
2x2x5x5x5
This doesn't match the engraved number, so it's out.

Pattern z→y→x:

3
2 3 5 5
2

↓ z rotation

2x3
2x2 3 3x5 5
2x5

↓ y rotation

2x3
2x2x3 3x3x5 3x5x5 2x2x5
2x5

↓ x rotation

2x3x3x3x5
2x2x3 2x3x3x5x5 3x5x5 2x2x2x3x5
2x2x2x5x5
This matches the engraved numbers, so it is a solution.

<Case2: α=5, β=3, γ=2>

Pattern x→y→x:

5
2 5 3 3
2

↓ x rotation

5x5
2 2x5 3 3x5
2x3
At this point, the U face does not match, so this is out.

Pattern x→z→x:

5
2 5 3 3
2

↓ x rotation

5x5
2 2x5 3 3x5
2x3
At this point, the U face does not match, so this is out.

Pattern y→z→x:

5
2 5 3 3
2

↓ y rotation

5
2x5 3x5 3x3 2x3
2
At this point, the L face does not match, so this is out.

Pattern z→y→x:

5
2 5 3 3
2

↓ z rotation

2x5
2x2 5 3x5 3
2x3

↓ y rotation

2x5
2x2x5 3x5x5 3x3x5 2x2x3
2x3
At this point, the L face does not match, so this is out.

<Conclusion>
1. Initial digits: α=3, β=5, γ=2

2. Calculation formula: new number = (original number) x (number of overlaps per 90-degree rotation of the cube)

3. 3-rotation pattern: z→y→x

4. The number calculated based on 1, 2, and 3 above matched the number engraved.

(Initial digits)
3
2 3 5 5
2

↓ z rotation

6
4 3 15 5
10

↓ y rotation

6
12 45 75 20
10

↓ x rotation

270
12 450 75 120
200

I'll contact Indy right away.

 

[Tito_H]

Later, I received a letter from Indy saying, "I turned the cube as you inferred, but nothing changed, and the whereabouts of the treasure are still unknown. However, I believe that the prime numbers (2, 3, 5) and the direction of rotation (z → y → x) have some meaning, so I am currently searching for further clues."

Good luck, Indy!

 

[MrCubert]

came for the cubes and left with a math problem

 

[Tito_H]

Who?