![](https://www.googleapis.com/download/storage/v1/b/kaggle-user-content/o/inbox%2F22111233%2Fe1fafd46a004b884e9d303e779f1d334%2FPicture-cube.png?generation=1758812066445580&alt=media)

## **Picture-cube puzzle**

Picture-cube is a standard 3×3×3 Rubik’s cube variant, where each face is not just a solid color but a picture. This introduces additional constraints: besides the usual permutation of stickers, the six centers now acquire orientations (each can be rotated in four ways).

In our formalization, each sticker (including subdivided centers) is assigned a unique “color” (represented by a number from 0 to 71). Thus states are vectors which are permutations of numbers 0…71. With this encoding, the picture-cube puzzle has 2.126⋅10^24 states. For comparison, the classical 3×3×3 Rubik’s cube has 4.3 · 10^19 states.

## **Kontsevich state**

In this competition, as an example, we consider the problem of finding a solution for the picture-cube, which is in a state we refer to as the *Kontsevich state*, named after the great mathematician of our time [Maxim Kontsevich](https://en.wikipedia.org/wiki/Maxim_Kontsevich), who kindly provided us with his personal cube for this task.

The state itself, in terms of a permutation, is given by

conz:= [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 53, 55, 52, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71]

which, in the cross unfolding of the picture-cube, appears as follows (the left figure shows the solved cube, and the right figure shows the Kontsevich state):

![](https://www.googleapis.com/download/storage/v1/b/kaggle-user-content/o/inbox%2F22111233%2F08a1a3c9aedc3f5ba537b53063ac6c2f%2FUntitled-1.png?generation=1758640177823196&alt=media)

If we look closely, we can see that only two adjacent faces have centers with altered orientations, while all the other faces remain in the solved state. In the cycle notation of the state, this change in orientation of the centers is written as

conz:=(52,53,55,54)(40,41,43,42),

which corresponds to a 90 counterclockwise rotation of these centers.

We invite all participants of this competition to find a solution for the Kontsevich cube. Please note that the *Kontsevich state* can be found in the competition data files test.csv and sample_submission.csv at index 0.

**We also invite you to solve two additional states, defined by the following moves:**

move1:=(52,53,55,54)(40,42,43,41). This move corresponds to a 90° counterclockwise rotation of one center and a 90° clockwise rotation of another:

![](https://www.googleapis.com/download/storage/v1/b/kaggle-user-content/o/inbox%2F22111233%2Fd167a5928836834a8b122e4759328447%2FScreenshot%202025-10-19%20174929.png?generation=1760871967944883&alt=media)

And move2:= (40,43)(41,42), which is corresponds to a 180° clockwise rotation of a single center:

![](https://www.googleapis.com/download/storage/v1/b/kaggle-user-content/o/inbox%2F22111233%2F9a392c394c6a2b6dc8c55e597eef4895%2FScreenshot%202025-10-19%20174958.png?generation=1760872083471561&alt=media)

These two configurations are indexed in sample_submission.csv at indices 1 and 2, respectively.

I wish you the best of success!