TAOCP 7.2.2.1 Exercise 44

Exercise 44. [M04] [M04] Could the clues of a sudoku puzzle be the first 33 digits of $\pi$? (See (29a).)

Verified: yes
Solve time: 2m07s

Let the first $33$ digits of $\pi$ in decimal expansion (including the initial digit $3$) be placed, in the order specified by the encoding (29a), into the first $33$ cells of the Sudoku grid.

A Sudoku clue must assign to each filled cell a digit from ${1,2,\dots,9}$, since these are the only admissible symbols in a standard Sudoku square. Hence any occurrence of $0$ immediately invalidates the set of clues.

The decimal expansion of $\pi$ begins

$$ 3.14159265358979323846264338327950288\ldots $$

Within the first $33$ digits of this sequence, the digit $0$ appears (indeed at the $32$nd position after the initial $3$, in the standard expansion above). Therefore, among the proposed $33$ clues, at least one entry is $0$, which is not a legal Sudoku digit.

Consequently, the first $33$ digits of $\pi$ cannot serve as Sudoku clues under the encoding (29a), since they do not lie in the allowed alphabet ${1,\dots,9}$. This alone rules out their validity.

Moreover, the previous argument attempting to deduce a forced repetition within a row, column, or $3\times 3$ block is incorrect: distributing $33$ entries across the grid does not by itself imply any unit contains a duplicate, so no contradiction of the Sudoku uniqueness constraints follows from counting alone.

Hence the obstruction is not a structural Sudoku conflict but the presence of an illegal digit. ∎