The Number of the Creased

Here’s an idea for a story à la M.R. James. A middle-aged scholar opens some mail one morning and finds nothing inside one envelope but a strip of paper with the numbers 216348597 written on it in sinister red ink. Someone has folded the strip several times so that there are creases between groups of numbers, like this: 216|348|5|97. Wondering what the significance of the creases is, the scholar hits on the step of adding the numbers created by them:


216 + 348 + 5 + 97 = 666

After that… Well, I haven’t written the story yet. But that beginning raises an obvious question. Is there any other way of getting a Number of the Creased from 216348597? That is, can you get 666, the Number of the Beast, by dividing 216348597 in another way? Yes, you can. In fact, there are six ways of creating 666 by dividing-and-summing 216348597:


666 = 2 + 1 + 634 + 8 + 5 + 9 + 7
666 = 2 + 163 + 485 + 9 + 7
666 = 216 + 348 + 5 + 97
666 = 21 + 63 + 485 + 97
666 = 21 + 6 + 34 + 8 + 597
666 = 2 + 16 + 3 + 48 + 597


216348597 is a permutation of 123456789, so does 123456789 yield a Number of the Creased? Yes. Two of them, in fact:


666 = 123 + 456 + 78 + 9
666 = 1 + 2 + 3 + 4 + 567 + 89


And 987654321 yields another:


666 = 9 + 87 + 6 + 543 + 21


And what about other permutations of 123456789? These are the successive records:

Using 123456789

666 = 123 + 456 + 78 + 9
666 = 1 + 2 + 3 + 4 + 567 + 89 (c=2)


Using 123564789

666 = 12 + 3 + 564 + 78 + 9
666 = 123 + 56 + 478 + 9
666 = 1 + 2 + 3 + 564 + 7 + 89 (c=3)


Using 125463978

666 = 1 + 2 + 5 + 4 + 639 + 7 + 8
666 = 12 + 546 + 3 + 97 + 8
666 = 1 + 254 + 6 + 397 + 8
666 = 1 + 2 + 546 + 39 + 78 (c=4)


Using 139462578

666 = 13 + 9 + 4 + 625 + 7 + 8
666 = 139 + 462 + 57 + 8
666 = 1 + 394 + 6 + 257 + 8
666 = 1 + 39 + 46 + 2 + 578
666 = 13 + 9 + 4 + 62 + 578 (c=5)


Using 216348597

666 = 2 + 1 + 634 + 8 + 5 + 9 + 7
666 = 2 + 163 + 485 + 9 + 7
666 = 216 + 348 + 5 + 97
666 = 21 + 63 + 485 + 97
666 = 21 + 6 + 34 + 8 + 597
666 = 2 + 16 + 3 + 48 + 597 (c=6)



216348597 is the best of the bestial. No other permutation of 123456789 yields as many as six Numbers of the Creased.