Henry Ernest Dudeney/Puzzles and Curious Problems/88 - Digital Progression/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $88$

Digital Progression

If you arrange the nine digits in three numbers thus, $147$, $258$, $369$,

they have a common difference of $111$ and are therefore in arithmetic progression.

Can you find $4$ ways of rearranging the $9$ digits so that in each case the number shall have a common difference,

and the middle number be in every case the same?

Solution

$297$, $564$, $831$

$291$, $564$, $837$

$237$, $564$, $891$

$231$, $564$, $897$

where the common differences are respectively $267$, $273$, $327$ and $333$.

Historical Note

Victor Meally informs Martin Gardner that subsequent to Dudeney's $4$ examples, Victor Michael Jean-Marie Thébault reported on $760$ such progressions in his Les Récréations Mathématiques from $1952$.

He also found that in addition to $456$ and its permutations, the middle number may be any of the permutations of $258$, $267$, $348$ and $357$.

Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $88$. -- Digital Progression
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $125$. Digital Progression