Entirely Ridiculously Big Numbers - Numberphile

The discussion focuses on calculating the number of possible scrambled states for various Rubik's Cubes, starting with a classic 3x3 cube and progressing to larger, even-numbered cubes. For a standard 3x3 Rubik's Cube, there are approximately 43 quintillion possible scrambles, or 4.3 x 10^19 states. This number represents all unique ways the cube can be mixed up, including the solved state, without overcounting identical states due to reflections or internal center piece arrangements. To understand how these numbers are derived, the explanation begins with a smaller, 2x2 cube. A 2x2 cube has eight "cubelets" (or "cubies"). The number of ways to arrange these cubelets in their eight positions is 8 factorial (8!), which is 8 x 7 x 6 x ... x 1. Additionally, each cubelet can be oriented in three different ways. However, once seven cubelets are oriented, the orientation of the eighth is automatically fixed. So, this contributes 3 to the power of 7 (3^7) possibilities. Multiplying these gives a large number, but this includes rotations of the entire cube that result in the same state. To correct for this, this number is divided by 24 (6 faces * 4 rotations per face), resulting in approximately 3.6 million unique states for a 2x2 cube. This initial calculation for the corners is referred to as 'C' for future calculations. Next, the calculation for a 4x4 cube is explored. A 4x4 cube introduces different types of pieces: corners, edges, and centers. Unlike the 2x2 cube where all pieces are equivalent, these types cannot be interchanged. The total number of states for a 4x4 cube is found by multiplying the possible arrangements of its corners, edges, and centers. The corner arrangements for a 4x4 cube are identical to the corner calculation for the 2x2 cube, so the number 'C' is used again. For the edges of a 4x4 cube, there are 12 edges, each with two cubelets, totaling 24 edge pieces. These 24 edge pieces can be arranged in 24 factorial (24!) ways. Unlike the corner pieces, the mechanics of the puzzle mean that once an edge piece is placed in a slot, its orientation is fixed, so there's no additional factor for rotation. This number, referred to as 'E', is approximately 6 x 10^23, or 620 sextillion. For the center pieces of a 4x4 cube, there are six faces, each with four center pieces, totaling 24 center pieces. Initially, this also seems to be 24 factorial. However, within each face, the four center pieces are identical (e.g., four white center pieces). Swapping these identical pieces does not create a new state. Since there are 4 factorial (4!) ways to arrange these four identical centers on one face, the initial 24 factorial must be divided by 4 factorial for each of the six faces. This means dividing by (4!)^6. This number, referred to as 'K', is approximately 3 x 10^15, or 3 quadrillion. To find the total number of states for the 4x4 cube, C, E, and K are multiplied together, and then this product is divided by 24 (for overall cube rotations). This yields approximately 7 x 10^45 possible states, a significant increase from the 3x3 cube's 4.3 x 10^19 states. The discussion then moves to a 6x6 cube. The corner calculation remains 'C'. For the edges, a 6x6 cube has two distinct types of edge pieces: central edges and "wing" edges. These types are constrained to their respective "corridors" and cannot be interchanged. The number of arrangements for the central edges is 'E' (as calculated for the 4x4 cube's edges), and the number of arrangements for the wing edges is also 'E'. Therefore, for the 6x6 cube, the edge contribution is E squared (E^2). For the centers of a 6x6 cube, there are four fundamentally different kinds of center pieces within each face. These different kinds of centers are also constrained to their own "corridors." For each of these four kinds of centers, the arrangement possibilities are 'K' (as calculated for the 4x4 cube's centers). Thus, the total center contribution is K to the power of 4 (K^4). Multiplying C, E^2, and K^4, and then dividing by 24, gives the total number of states for a 6x6 cube, which is approximately 1.6 x 10^116. This number is described as being "beyond atoms in the universe." Finally, a general algorithm is presented for even-numbered cubes, represented as '2n' cubes (where 'n' is half the width). For a 2n cube: - The corner contribution remains 'C'. - The edge contribution becomes E to the power of (n-1) [E^(n-1)], because there are (n-1) fundamentally different types of edge pieces. - The center contribution becomes K to the power of (n-1) squared [K^((n-1)^2)], because there are (n-1)^2 fundamentally different types of center pieces. - The entire product is then divided by 24. Applying this algorithm to a 10x10 cube (where n=5): - Corners: C - Edges: E^(5-1) = E^4 - Centers: K^((5-1)^2) = K^(4^2) = K^16 - The total is (C * E^4 * K^16) / 24. This calculation for a 10x10 cube results in approximately 10^349 possible states, an "entirely ridiculously big number" that far surpasses even astronomical scales. The significant increase in magnitude is due to the exponential nature of the center and edge calculations, particularly the K to the power of n-squared term. The video concludes by mentioning that calculations for odd-numbered cubes are more complex and are discussed in an extended video.