**From New Scientist #2352, 20th July 2002** [link]

The septitude of a two-digit number, *n*, is obtained by first dividing *n* by 7 and noting the remainder, *r*. We use *r* to calculate a new number as follows:

if *r* is 0, 1, 2, 3, 4, 5, 6 then the new number is *n* + 29, *n* – 23, *n* + 15, *n* – 40, *n* + 31, *n* + 25, *n* – 37 respectively.

If the new number is in the range 10 to 99 then it is the septitude of *n*, otherwise add or subtract 84 from the new number to get a number in the range 10 to 99, and that latter number is the septitude of *n*.

For example, the septitudes of 14, 15, and 74 are 43, 76, and 21 respectively.

In the crossnumber puzzle (below):

each of the four answers, 1 Across, 3 Across, 1 Down and 2 Down,
equals the septitude of another of these four answers. Also 3 Across is larger than 2 Down.

What are 1 Across and 3 Across?

[enigma1196]

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This Python program runs in 58ms.

Solution:The solution to 1 across is 63. The solution to 3 across is 92.By repeatedly calculating “septitudes” we get the following cyclical chain: 63 → 92 → 69 → 32 → 63.