Henry Ernest Dudeney/Modern Puzzles/102 - The Nine Barrels

Modern Puzzles by Henry Ernest Dudeney: $102$

The Nine Barrels

In how many different ways may these nine barrels be arranged in three tiers of three

so that no barrel shall have a smaller number than its own below it or to the right of it?

The first correct arrangement that will occur to you is $1 \ 2 \ 3$ at the top then $4 \ 5 \ 6$ in the second row, and $7 \ 8 \ 9$ at the bottom,

and my sketch gives a second arrangement.

How many are there altogether?

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Arithmetical and Algebraical Problems: Various Arithmetical and Algebraical Problems: $102$. -- The Nine Barrels
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $138$. The Nine Barrels