TAOCP 7.2.2.1 Exercise 289
Please provide Figure (36) and the full image for exercise 289(c), or the corresponding region coordinates.
Section 7.2.2.1: Dancing Links
Exercise 289. ▶ [29] [29] (Pentominoes of pentominoes.) Magnify the $3 \times 20$ pentomino packing (36) by replacing each of its unit cells by (a) $3 \times 4$ rectangles; (b) $4 \times 3$ rectangles. In how many ways can the resulting 720-cell shape be packed with twelve complete sets of twelve pentominoes, using one set for each of the original pentomino regions?
(c) Also partition the 720-cell shape below into $3 \times 20$ approximately square 12-cell regions, by assigning each gray cell to an adjacent region. (This shape has been superimposed on a grid whose $\sqrt{12} \times \sqrt{12}$ regions are perfectly square.) Minimize the total perimeter of the 60 resulting regions, and try for a pleasantly symmetrical solution.
Use your partition to present a scaled-up version of (36), again with 12 complete sets.
Verified: no
Solve time: 1m50s
Please provide Figure (36) and the full image for exercise 289(c), or the corresponding region coordinates. With those, I can write the requested solution in the specified format:
## Setup## Solution## Verification## Notes
with a complete derivation of the packing counts, a proof of the partition perimeter minimum for part (c), and the final boxed answers.