Entirely Ridiculously Big Numbers - Numberphile

Key Concepts

• Cubelet: The individual smaller cubes that make up a larger Rubik’s Cube.
• State: A specific configuration or "scramble" of a cube.
• Factorial ($n!$): The product of all positive integers up to $n$ (e.g., $4! = 4 \times 3 \times 2 \times 1 = 24$).
• Even-width Cubes: Cubes with an even number of cubelets per edge (e.g., 2x2, 4x4, 6x6, 10x10).
• Combinatorics: The branch of mathematics used to calculate the number of possible arrangements of cubelets.
• Exponential Growth: The rapid increase in the number of states as the cube size increases, specifically related to $n^2$.

---

1. Calculating States for Even-Width Cubes

The video establishes a mathematical framework for determining the number of possible states for even-width cubes ($2n$). The total number of states is derived by calculating the arrangements of three distinct types of pieces: Corners, Edges, and Centers.

The 2x2 Cube (The Base Case)

• Corners: There are 8 corner pieces. The number of permutations is $8!$.
• Rotation: Each corner can be oriented in 3 ways. Because the orientation of the final corner is constrained by the others, the multiplier is $3^7$ rather than $3^8$.
• Symmetry: To account for the fact that the cube can be rotated in space (24 possible orientations), the total is divided by 24.
• Result: Approximately 3.6 million states.

The 4x4 Cube

• Corners ($C$): Same as the 2x2 cube.
• Edges ($E$): There are 24 edge pieces. Since they are distinct and their orientation is fixed by the mechanics, the number of arrangements is $24!$ (approx. $6 \times 10^{23}$).
• Centers ($K$): There are 24 center pieces. Because center pieces of the same color are identical, we must divide by $4!$ (24) for each of the 6 colors.
• Total Formula: $\frac{C \times E \times K}{24} \approx 7 \times 10^{45}$ states.

The 6x6 and 10x10 Cubes

• Corridor Structure: As cubes get larger, pieces are restricted to "corridors." For example, in a 6x6, edge pieces are split into "central edges" and "wing edges," which cannot swap positions.
• Generalization for $2n$ Cubes:
  • Edges: $E^{n-1}$
  • Centers: $K^{(n-1)^2}$
• The 10x10 Result: By plugging $n=5$ into the generalized formula, the number of states reaches approximately $10^{349}$. The speaker notes this number exceeds the number of atoms in the universe, even if every atom were replaced by an entire universe, repeated four times.

---

2. Key Arguments and Perspectives

• Overcounting: A critical part of the methodology is identifying identical states. If swapping two pieces results in an indistinguishable visual state, those arrangements must be divided out to avoid overcounting.
• The "Formality" of Division: The speaker notes that as the total number of states reaches astronomical levels, the final division by 24 (to account for cube rotation) becomes mathematically negligible, though it remains a formal requirement for accuracy.
• Constraint Mechanics: The physical construction of the cube dictates the math. For instance, the fact that center pieces of the same color are indistinguishable is a physical property that drastically reduces the total state space compared to if every piece were unique.

---

3. Notable Quotes

• "There’s more possible scrambles of this puzzle than there are atoms in the universe." — Regarding the 6x6 cube.
• "It’s exponential in $n$, but it’s not actually just exponential in $n$, it’s exponential in $n^2$. And that’s why it gets so big." — Explaining the rapid growth of states in larger cubes.

---

4. Synthesis and Conclusion

The complexity of a Rubik's Cube's state space is determined by the number of permutations of its constituent parts (corners, edges, and centers) adjusted for physical constraints and indistinguishable pieces. While a standard 3x3 cube has 43 quintillion states, the state space grows at an exponential rate relative to the square of the cube's width. By the time one reaches a 10x10 cube, the number of possible configurations ($10^{349}$) surpasses human comprehension, illustrating how simple mechanical rules can generate near-infinite complexity. For those interested in the more complex mathematics of odd-numbered cubes, the speaker directs viewers to the "Numberphile 2" channel.