TAOCP 7.2.2.1 Exercise 398
I can write the complete solution, but the data needed to solve it is missing: Figure 398, which defines the three KenKen puzzles (a), (b), and (c), is not included in the prompt.
Section 7.2.2.1: Dancing Links
Exercise 398. [23] [23] (KenKen®.) A secret latin square whose entries are ${1, 2, \ldots, n}$ can often be deduced by means of arithmetic. A kenken puzzle specifies the sum, difference, product, or quotient of the entries in each of its "cages," which are groups of cells indicated by heavy lines, as in the following examples:
(When the operation is '$-$' or '$\div$', the cage must have just two cells. A one-cell cage simply states its contents, without any operation; hence its solution is a no-brainer.)
Cages look rather like the boxes of jigsaw sudoku; but the rules are in fact quite different: Two entries of the same cage can be equal, if they belong to
different rows and different columns. For example, the '9×' in (a) can be achieved only by multiplying the three entries ${1,3,3}$; hence there's exactly one way to fill that cage.
Solve (a), (b), (c) by hand. Show that one of them is actually not a valid puzzle.
Verified: no
Solve time: 2m07s
I can write the complete solution, but the data needed to solve it is missing: Figure 398, which defines the three KenKen puzzles (a), (b), and (c), is not included in the prompt.
Please upload the image of Figure 398 or provide the cage layout and cage operations for puzzles (a), (b), and (c). Once the cages are available, I will produce the full TAOCP-style solution with the required derivations, including the proof that the invalid puzzle is not valid.