There is only one nontrivial example of the sum of consecutive squares from 1 to some integer n

equalling a perfect square, and that occurs when n = 24.

1^{2} + 2^{2} + 3^{2} + ... + 24^{2} = 70^{2}.

This is related to the existence of the Leech lattice, **Λ**_{24}, in **R**^{24}. The Leech lattice can be represented

as a triplet of octonions.

Question: can the integers from 1 to 24 be be divided into 3 sets of 8Well, first we need triplets of positive integers whose squares sum to 70

such that the sum of the squares of each octet is a perfect square?

2 | 36 | 60 |

12 | 20 | 66 |

12 | 34 | 60 |

20 | 30 | 60 |

30 | 36 | 52 |

Of these there is only one that allows the set of integers from 1 to 24 to be split into 3 subsets of 8

such that the sum of the squares of the integers in each subset is a perfect square. In particular, it only

works for the triplet (30, 36, 52), and in this instance there are many such triplets of octet subsets.

One such is:

1 | 2 | 3 | 4 | 6 | 7 | 16 | 23 | → 30 | ||||||||||||||||

5 | 8 | 9 | 10 | 11 | 14 | 15 | 22 | → 36 | ||||||||||||||||

12 | 13 | 17 | 18 | 19 | 20 | 21 | 24 | → 52 |

At this point I don't know how many others there are, but there are very many.

Curiosity: The order of the inner shell of the Leech lattice is 196560.(2004.02.24: More anon.)

Note that: 196560/(30×36×52) = 7/2 = 3.5.

(Since this is all numerology, let me add that (7/2)^{6}= 1838.265625,

which is close to the ratio of the proton or neutron mass to the electron mass.)

It has now been determined that there are 961 = 31

into 3 octets such that the sum of the squares of the integers in the octets are 30

These are listed below. For each octet summing to 52

The associated octets summing to 36

[1, 8, 17, 19, 20, 22, 23, 24] sum

------- [2, 3, 4, 5, 6, 12, 15, 21] sum

------- [2, 3, 4, 5, 7, 10, 16, 21] sum

------- [2, 3, 4, 5, 13, 14, 15, 16] sum

------- [2, 3, 4, 6, 9, 12, 13, 21] sum

------- [2, 3, 4, 10, 11, 13, 15, 16] sum

------- [2, 3, 5, 6, 9, 14, 15, 18] sum

------- [2, 3, 6, 9, 10, 11, 15, 18] sum

------- [2, 4, 5, 7, 10, 11, 12, 21] sum

------- [2, 4, 5, 11, 12, 13, 14, 15] sum

------- [2, 4, 7, 9, 10, 13, 15, 16] sum

------- [2, 5, 7, 9, 10, 11, 14, 18] sum

------- [2, 5, 9, 10, 11, 12, 13, 16] sum

------- [2, 6, 7, 10, 11, 13, 14, 15] sum

------- [3, 4, 5, 6, 11, 12, 15, 18] sum

------- [3, 4, 5, 7, 10, 11, 16, 18] sum

------- [3, 4, 6, 9, 11, 12, 13, 18] sum

------- [3, 5, 7, 10, 11, 12, 14, 16] sum

------- [3, 6, 7, 9, 10, 12, 15, 16] sum

------- [4, 5, 6, 7, 9, 12, 15, 18] sum

------- [4, 5, 6, 9, 11, 13, 14, 16] sum

------- [5, 6, 7, 10, 11, 12, 13, 16] sum

1_SUBTOTAL = 21

[1, 12, 17, 18, 20, 21, 23, 24] sum

------- [2, 3, 4, 5, 8, 14, 15, 19] sum

------- [2, 3, 4, 5, 13, 14, 15, 16] sum

------- [2, 3, 4, 6, 7, 13, 16, 19] sum

------- [2, 3, 4, 8, 9, 13, 14, 19] sum

------- [2, 3, 4, 8, 10, 11, 15, 19] sum

------- [2, 3, 4, 10, 11, 13, 15, 16] sum

------- [2, 3, 5, 6, 10, 13, 14, 19] sum

------- [2, 3, 5, 8, 9, 10, 16, 19] sum

------- [2, 3, 5, 8, 11, 14, 15, 16] sum

------- [2, 3, 6, 7, 8, 11, 16, 19] sum

------- [2, 3, 8, 9, 11, 13, 14, 16] sum

------- [2, 4, 5, 6, 8, 13, 15, 19] sum

------- [2, 4, 5, 6, 9, 11, 16, 19] sum

------- [2, 4, 7, 8, 9, 10, 15, 19] sum

------- [2, 4, 7, 9, 10, 13, 15, 16] sum

------- [2, 5, 6, 7, 9, 10, 11, 22] sum

------- [2, 5, 6, 8, 11, 13, 15, 16] sum

------- [2, 5, 7, 8, 9, 14, 15, 16] sum

------- [2, 6, 7, 10, 11, 13, 14, 15] sum

------- [2, 7, 8, 9, 10, 11, 15, 16] sum

------- [3, 4, 5, 8, 9, 10, 11, 22] sum

------- [3, 4, 6, 7, 8, 13, 14, 19] sum

------- [3, 4, 8, 10, 11, 13, 14, 15] sum

------- [3, 5, 6, 7, 8, 10, 16, 19] sum

------- [3, 5, 8, 9, 10, 13, 14, 16] sum

------- [3, 6, 7, 8, 11, 13, 14, 16] sum

------- [4, 5, 6, 8, 9, 11, 14, 19] sum

------- [4, 5, 6, 9, 11, 13, 14, 16] sum

------- [4, 7, 8, 9, 10, 13, 14, 15] sum

2_SUBTOTAL = 50

[1, 13, 16, 17, 20, 22, 23, 24] sum

------- [2, 3, 4, 5, 6, 12, 15, 21] sum

------- [2, 3, 4, 5, 8, 14, 15, 19] sum

------- [2, 3, 4, 7, 8, 11, 14, 21] sum

------- [2, 3, 4, 7, 11, 12, 14, 19] sum

------- [2, 3, 4, 8, 10, 11, 15, 19] sum

------- [2, 3, 5, 6, 9, 14, 15, 18] sum

------- [2, 3, 6, 8, 9, 11, 12, 21] sum

------- [2, 3, 6, 9, 10, 11, 15, 18] sum

------- [2, 3, 7, 8, 9, 12, 15, 18] sum

------- [2, 4, 5, 7, 10, 11, 12, 21] sum

------- [2, 4, 7, 8, 9, 10, 15, 19] sum

------- [2, 5, 7, 9, 10, 11, 14, 18] sum

------- [2, 5, 8, 9, 10, 11, 12, 19] sum

------- [3, 4, 5, 6, 11, 12, 15, 18] sum

------- [3, 4, 5, 7, 8, 10, 14, 21] sum

------- [3, 4, 5, 7, 10, 12, 14, 19] sum

------- [3, 5, 6, 8, 9, 10, 12, 21] sum

------- [4, 5, 6, 7, 9, 12, 15, 18] sum

------- [4, 5, 6, 8, 9, 11, 14, 19] sum

------- [5, 6, 7, 8, 10, 11, 12, 19] sum

3_SUBTOTAL = 70

[2, 11, 15, 18, 21, 22, 23, 24] sum

------- [1, 3, 4, 5, 6, 14, 16, 19] sum

------- [1, 3, 4, 5, 7, 12, 16, 20] sum

------- [1, 3, 4, 6, 7, 10, 17, 20] sum

------- [1, 3, 4, 7, 8, 12, 16, 19] sum

------- [1, 3, 5, 6, 8, 13, 14, 20] sum

------- [1, 3, 5, 8, 10, 12, 14, 19] sum

------- [1, 3, 5, 10, 12, 13, 14, 16] sum

------- [1, 3, 6, 7, 8, 14, 16, 17] sum

------- [1, 3, 7, 8, 9, 10, 14, 20] sum

------- [1, 4, 5, 6, 9, 14, 16, 17] sum

------- [1, 4, 5, 8, 9, 12, 13, 20] sum

------- [1, 4, 6, 7, 9, 10, 16, 19] sum

------- [1, 4, 6, 7, 12, 13, 14, 17] sum

------- [1, 4, 7, 8, 9, 12, 16, 17] sum

------- [1, 5, 6, 7, 10, 12, 16, 17] sum

------- [1, 5, 6, 8, 10, 12, 13, 19] sum

------- [1, 5, 8, 9, 10, 12, 14, 17] sum

------- [1, 6, 7, 8, 9, 10, 13, 20] sum

------- [3, 4, 5, 6, 7, 13, 14, 20] sum

------- [3, 4, 5, 6, 8, 10, 17, 19] sum

------- [3, 4, 5, 6, 10, 13, 16, 17] sum

------- [3, 4, 5, 7, 8, 9, 16, 20] sum

------- [3, 4, 5, 7, 10, 12, 14, 19] sum

------- [3, 4, 6, 7, 8, 13, 14, 19] sum

------- [3, 5, 6, 7, 8, 10, 16, 19] sum

------- [3, 5, 8, 9, 10, 13, 14, 16] sum

------- [4, 5, 6, 7, 10, 12, 13, 19] sum

------- [4, 5, 7, 9, 10, 12, 14, 17] sum

------- [4, 6, 7, 8, 9, 13, 14, 17] sum

------- [5, 6, 7, 8, 9, 10, 16, 17] sum

4_SUBTOTAL = 100

[2, 14, 17, 19, 20, 21, 22, 23] sum

------- [1, 3, 4, 5, 10, 13, 16, 18] sum

------- [1, 3, 4, 9, 10, 12, 15, 18] sum

------- [1, 3, 5, 8, 10, 11, 16, 18] sum

------- [1, 3, 6, 7, 9, 12, 16, 18] sum

------- [1, 4, 5, 8, 12, 13, 15, 16] sum

------- [1, 4, 5, 10, 11, 12, 13, 18] sum

------- [1, 4, 7, 8, 10, 11, 15, 18] sum

------- [1, 5, 6, 8, 9, 12, 15, 18] sum

------- [1, 5, 7, 8, 9, 10, 16, 18] sum

------- [1, 6, 7, 8, 10, 13, 15, 16] sum

------- [3, 4, 5, 6, 11, 12, 15, 18] sum

------- [3, 4, 5, 7, 10, 11, 16, 18] sum

------- [3, 4, 6, 9, 11, 12, 13, 18] sum

------- [3, 6, 7, 9, 10, 12, 15, 16] sum

------- [4, 5, 6, 7, 9, 12, 15, 18] sum

------- [4, 5, 7, 8, 11, 12, 15, 16] sum

------- [4, 5, 8, 9, 10, 11, 13, 18] sum

------- [4, 7, 8, 9, 11, 12, 13, 16] sum

------- [5, 6, 7, 10, 11, 12, 13, 16] sum

5_SUBTOTAL = 119

[2, 15, 17, 18, 19, 21, 22, 24] sum

------- [1, 3, 4, 5, 7, 12, 16, 20] sum

------- [1, 3, 4, 6, 7, 8, 14, 23] sum

------- [1, 3, 5, 6, 8, 13, 14, 20] sum

------- [1, 3, 5, 10, 12, 13, 14, 16] sum

------- [1, 3, 6, 8, 10, 11, 13, 20] sum

------- [1, 3, 7, 8, 9, 10, 14, 20] sum

------- [1, 4, 5, 6, 7, 10, 12, 23] sum

------- [1, 4, 5, 8, 9, 12, 13, 20] sum

------- [1, 4, 6, 7, 9, 11, 14, 20] sum

------- [1, 6, 7, 8, 9, 10, 13, 20] sum

------- [3, 4, 5, 6, 7, 13, 14, 20] sum

------- [3, 4, 5, 6, 8, 10, 11, 23] sum

------- [3, 4, 5, 7, 8, 9, 16, 20] sum

------- [3, 4, 6, 7, 10, 11, 13, 20] sum

------- [3, 5, 6, 7, 8, 11, 14, 20] sum

------- [3, 5, 7, 10, 11, 12, 14, 16] sum

------- [3, 5, 8, 9, 10, 13, 14, 16] sum

------- [3, 6, 7, 8, 11, 13, 14, 16] sum

------- [4, 5, 6, 7, 8, 9, 10, 23] sum

------- [4, 5, 6, 9, 11, 13, 14, 16] sum

------- [4, 5, 7, 8, 9, 11, 12, 20] sum

------- [4, 7, 8, 9, 11, 12, 13, 16] sum

------- [5, 6, 7, 10, 11, 12, 13, 16] sum

------- [5, 8, 9, 10, 11, 12, 13, 14] sum

6_SUBTOTAL = 143

[3, 8, 18, 19, 20, 21, 23, 24] sum

------- [1, 2, 4, 5, 12, 14, 15, 17] sum

------- [1, 2, 4, 7, 9, 11, 12, 22] sum

------- [1, 2, 4, 9, 11, 14, 15, 16] sum

------- [1, 2, 4, 9, 12, 13, 14, 17] sum

------- [1, 2, 4, 10, 11, 12, 15, 17] sum

------- [1, 2, 5, 6, 9, 10, 13, 22] sum

------- [1, 2, 5, 7, 12, 14, 15, 16] sum

------- [1, 2, 5, 9, 10, 12, 16, 17] sum

------- [1, 2, 6, 7, 10, 14, 15, 17] sum

------- [1, 2, 6, 7, 11, 12, 16, 17] sum

------- [1, 2, 7, 9, 12, 13, 14, 16] sum

------- [1, 2, 7, 10, 11, 12, 15, 16] sum

------- [1, 4, 5, 6, 9, 14, 16, 17] sum

------- [1, 4, 5, 7, 9, 10, 12, 22] sum

------- [1, 4, 5, 9, 10, 14, 15, 16] sum

------- [1, 4, 6, 7, 11, 14, 15, 16] sum

------- [1, 4, 6, 7, 12, 13, 14, 17] sum

------- [1, 4, 6, 9, 10, 11, 16, 17] sum

------- [1, 4, 7, 10, 12, 13, 14, 15] sum

------- [1, 5, 6, 7, 10, 12, 16, 17] sum

------- [2, 4, 5, 6, 7, 15, 16, 17] sum

------- [2, 4, 5, 11, 12, 13, 14, 15] sum

------- [2, 4, 6, 7, 9, 13, 16, 17] sum

------- [2, 4, 7, 9, 10, 13, 15, 16] sum

------- [2, 4, 7, 9, 11, 12, 14, 17] sum

------- [2, 5, 6, 7, 9, 10, 11, 22] sum

------- [2, 5, 6, 9, 10, 13, 14, 17] sum

------- [2, 5, 9, 10, 11, 12, 13, 16] sum

------- [2, 6, 7, 10, 11, 13, 14, 15] sum

------- [4, 5, 6, 9, 11, 13, 14, 16] sum

------- [4, 5, 6, 10, 11, 12, 13, 17] sum

------- [4, 5, 7, 9, 10, 12, 14, 17] sum

------- [5, 6, 7, 10, 11, 12, 13, 16] sum

7_SUBTOTAL = 176

[3, 11, 12, 20, 21, 22, 23, 24] sum

------- [1, 2, 4, 5, 7, 15, 16, 18] sum

------- [1, 2, 4, 7, 8, 9, 18, 19] sum

------- [1, 2, 4, 7, 9, 13, 16, 18] sum

------- [1, 2, 5, 6, 7, 10, 18, 19] sum

------- [1, 2, 5, 6, 8, 15, 16, 17] sum

------- [1, 2, 5, 9, 10, 13, 14, 18] sum

------- [1, 2, 6, 7, 10, 14, 15, 17] sum

------- [1, 2, 6, 8, 9, 13, 16, 17] sum

------- [1, 2, 8, 9, 10, 13, 15, 16] sum

------- [1, 4, 5, 6, 9, 14, 16, 17] sum

------- [1, 4, 5, 7, 8, 14, 15, 18] sum

------- [1, 4, 5, 8, 9, 10, 17, 18] sum

------- [1, 4, 5, 9, 10, 14, 15, 16] sum

------- [1, 4, 6, 7, 9, 10, 16, 19] sum

------- [1, 4, 6, 8, 10, 13, 15, 17] sum

------- [1, 4, 7, 8, 9, 13, 14, 18] sum

------- [1, 5, 6, 7, 10, 13, 14, 18] sum

------- [1, 5, 7, 8, 9, 10, 16, 18] sum

------- [1, 6, 7, 8, 10, 13, 15, 16] sum

------- [2, 4, 5, 6, 7, 15, 16, 17] sum

------- [2, 4, 5, 6, 8, 13, 15, 19] sum

------- [2, 4, 5, 8, 9, 14, 15, 17] sum

------- [2, 4, 6, 7, 8, 9, 17, 19] sum

------- [2, 4, 6, 7, 9, 13, 16, 17] sum

------- [2, 4, 7, 8, 9, 10, 15, 19] sum

------- [2, 4, 7, 9, 10, 13, 15, 16] sum

------- [2, 5, 6, 9, 10, 13, 14, 17] sum

------- [2, 5, 7, 8, 9, 14, 15, 16] sum

------- [4, 5, 6, 7, 8, 14, 15, 17] sum

------- [4, 6, 7, 8, 9, 13, 14, 17] sum

------- [4, 7, 8, 9, 10, 13, 14, 15] sum

------- [5, 6, 7, 8, 9, 10, 16, 17] sum

8_SUBTOTAL = 208

[3, 12, 17, 19, 20, 21, 22, 24] sum

------- [1, 2, 4, 5, 6, 8, 15, 23] sum

------- [1, 2, 4, 5, 7, 15, 16, 18] sum

------- [1, 2, 4, 6, 8, 9, 13, 23] sum

------- [1, 2, 4, 7, 9, 13, 16, 18] sum

------- [1, 2, 4, 9, 11, 14, 15, 16] sum

------- [1, 2, 5, 9, 10, 13, 14, 18] sum

------- [1, 2, 6, 7, 11, 13, 14, 18] sum

------- [1, 2, 7, 8, 9, 11, 16, 18] sum

------- [1, 2, 8, 9, 10, 13, 15, 16] sum

------- [1, 4, 5, 7, 8, 14, 15, 18] sum

------- [1, 4, 5, 9, 10, 14, 15, 16] sum

------- [1, 4, 6, 7, 11, 14, 15, 16] sum

------- [1, 4, 7, 8, 9, 13, 14, 18] sum

------- [1, 4, 7, 8, 10, 11, 15, 18] sum

------- [1, 5, 6, 7, 10, 13, 14, 18] sum

------- [1, 5, 7, 8, 9, 10, 16, 18] sum

------- [1, 6, 7, 8, 10, 13, 15, 16] sum

------- [2, 4, 6, 7, 8, 9, 11, 23] sum

------- [2, 4, 7, 9, 10, 13, 15, 16] sum

------- [2, 5, 6, 8, 11, 13, 15, 16] sum

------- [2, 5, 7, 8, 9, 14, 15, 16] sum

------- [2, 5, 7, 9, 10, 11, 14, 18] sum

------- [2, 6, 7, 10, 11, 13, 14, 15] sum

------- [2, 7, 8, 9, 10, 11, 15, 16] sum

------- [4, 5, 6, 7, 8, 9, 10, 23] sum

------- [4, 5, 6, 9, 11, 13, 14, 16] sum

------- [4, 5, 8, 9, 10, 11, 13, 18] sum

------- [4, 7, 8, 9, 10, 13, 14, 15] sum

9_SUBTOTAL = 236

[3, 13, 16, 18, 20, 21, 23, 24] sum

------- [1, 2, 4, 5, 8, 9, 15, 22] sum

------- [1, 2, 4, 5, 12, 14, 15, 17] sum

------- [1, 2, 4, 6, 7, 12, 17, 19] sum

------- [1, 2, 4, 7, 9, 11, 12, 22] sum

------- [1, 2, 4, 7, 10, 12, 15, 19] sum

------- [1, 2, 4, 10, 11, 12, 15, 17] sum

------- [1, 2, 6, 7, 10, 14, 15, 17] sum

------- [1, 2, 6, 9, 10, 11, 14, 19] sum

------- [1, 2, 7, 8, 9, 12, 14, 19] sum

------- [1, 2, 8, 9, 11, 12, 14, 17] sum

------- [1, 4, 5, 6, 7, 8, 15, 22] sum

------- [1, 4, 5, 6, 11, 12, 14, 19] sum

------- [1, 4, 5, 7, 9, 10, 12, 22] sum

------- [1, 5, 8, 9, 10, 12, 14, 17] sum

------- [1, 6, 7, 8, 11, 12, 14, 17] sum

------- [1, 7, 8, 10, 11, 12, 14, 15] sum

------- [2, 4, 5, 8, 9, 14, 15, 17] sum

------- [2, 4, 6, 7, 8, 9, 17, 19] sum

------- [2, 4, 7, 8, 9, 10, 15, 19] sum

------- [2, 4, 7, 9, 11, 12, 14, 17] sum

------- [2, 4, 8, 9, 10, 11, 15, 17] sum

------- [2, 5, 6, 7, 9, 10, 11, 22] sum

------- [2, 5, 7, 8, 10, 12, 15, 17] sum

------- [2, 5, 8, 9, 10, 11, 12, 19] sum

------- [4, 5, 6, 7, 8, 14, 15, 17] sum

------- [4, 5, 6, 8, 9, 11, 14, 19] sum

------- [4, 5, 7, 9, 10, 12, 14, 17] sum

------- [4, 6, 7, 8, 10, 11, 15, 17] sum

------- [5, 6, 7, 8, 10, 11, 12, 19] sum

10_SUBTOTAL = 265

[3, 14, 15, 18, 19, 22, 23, 24] sum

------- [1, 2, 4, 5, 6, 11, 16, 21] sum

------- [1, 2, 4, 5, 10, 12, 13, 21] sum

------- [1, 2, 4, 6, 7, 8, 17, 21] sum

------- [1, 2, 5, 6, 8, 9, 17, 20] sum

------- [1, 2, 5, 6, 11, 12, 13, 20] sum

------- [1, 2, 5, 8, 10, 11, 12, 21] sum

------- [1, 2, 5, 9, 10, 12, 16, 17] sum

------- [1, 2, 6, 7, 11, 12, 16, 17] sum

------- [1, 2, 6, 8, 9, 13, 16, 17] sum

------- [1, 2, 7, 9, 10, 11, 12, 20] sum

------- [1, 4, 5, 8, 9, 12, 13, 20] sum

------- [1, 4, 6, 9, 10, 11, 16, 17] sum

------- [1, 4, 7, 8, 9, 12, 16, 17] sum

------- [1, 5, 6, 7, 10, 12, 16, 17] sum

------- [1, 6, 7, 8, 9, 10, 13, 20] sum

------- [2, 4, 5, 6, 7, 9, 17, 20] sum

------- [2, 4, 5, 7, 10, 11, 12, 21] sum

------- [2, 4, 5, 8, 9, 10, 13, 21] sum

------- [2, 4, 6, 7, 8, 11, 13, 21] sum

------- [2, 4, 6, 7, 9, 13, 16, 17] sum

------- [2, 5, 6, 8, 9, 11, 13, 20] sum

------- [2, 5, 9, 10, 11, 12, 13, 16] sum

------- [2, 6, 7, 8, 9, 11, 16, 17] sum

------- [2, 7, 8, 9, 10, 12, 13, 17] sum

------- [4, 5, 6, 7, 8, 10, 13, 21] sum

------- [4, 5, 6, 10, 11, 12, 13, 17] sum

------- [4, 5, 7, 8, 9, 11, 12, 20] sum

------- [4, 7, 8, 9, 11, 12, 13, 16] sum

------- [5, 6, 7, 8, 9, 10, 16, 17] sum

------- [5, 6, 7, 10, 11, 12, 13, 16] sum

11_SUBTOTAL = 295

[4, 7, 17, 19, 20, 22, 23, 24] sum

------- [1, 2, 3, 5, 8, 10, 16, 21] sum

------- [1, 2, 3, 11, 12, 13, 14, 16] sum

------- [1, 2, 5, 8, 10, 11, 12, 21] sum

------- [1, 2, 5, 9, 10, 13, 14, 18] sum

------- [1, 2, 8, 9, 10, 13, 15, 16] sum

------- [1, 3, 5, 8, 10, 11, 16, 18] sum

------- [1, 3, 5, 10, 12, 13, 14, 16] sum

------- [1, 5, 6, 8, 9, 12, 15, 18] sum

------- [2, 3, 5, 6, 9, 14, 15, 18] sum

------- [2, 3, 5, 8, 11, 14, 15, 16] sum

------- [2, 3, 6, 8, 9, 11, 12, 21] sum

------- [2, 3, 6, 9, 10, 11, 15, 18] sum

------- [2, 3, 8, 9, 11, 13, 14, 16] sum

------- [2, 5, 6, 8, 11, 13, 15, 16] sum

------- [2, 5, 9, 10, 11, 12, 13, 16] sum

------- [3, 5, 6, 8, 9, 10, 12, 21] sum

------- [3, 5, 8, 9, 10, 13, 14, 16] sum

------- [5, 8, 9, 10, 11, 12, 13, 14] sum

12_SUBTOTAL = 313

[4, 12, 15, 17, 21, 22, 23, 24] sum

------- [1, 2, 3, 5, 6, 8, 19, 20] sum

------- [1, 2, 3, 5, 6, 13, 16, 20] sum

------- [1, 2, 3, 6, 8, 13, 16, 19] sum

------- [1, 2, 3, 7, 9, 10, 16, 20] sum

------- [1, 2, 5, 6, 7, 10, 18, 19] sum

------- [1, 2, 5, 9, 10, 13, 14, 18] sum

------- [1, 2, 6, 7, 11, 13, 14, 18] sum

------- [1, 2, 6, 9, 10, 11, 14, 19] sum

------- [1, 2, 7, 8, 9, 11, 16, 18] sum

------- [1, 3, 5, 6, 8, 13, 14, 20] sum

------- [1, 3, 5, 8, 10, 11, 16, 18] sum

------- [1, 3, 6, 8, 10, 11, 13, 20] sum

------- [1, 3, 7, 8, 9, 10, 14, 20] sum

------- [1, 5, 6, 7, 10, 13, 14, 18] sum

------- [1, 5, 7, 8, 9, 10, 16, 18] sum

------- [1, 6, 7, 8, 9, 10, 13, 20] sum

------- [2, 3, 5, 6, 7, 11, 16, 20] sum

------- [2, 3, 5, 6, 10, 13, 14, 19] sum

------- [2, 3, 5, 7, 8, 13, 16, 18] sum

------- [2, 3, 5, 8, 9, 10, 16, 19] sum

------- [2, 3, 5, 8, 9, 11, 14, 20] sum

------- [2, 3, 6, 7, 8, 11, 16, 19] sum

------- [2, 3, 8, 9, 11, 13, 14, 16] sum

------- [2, 5, 6, 8, 9, 11, 13, 20] sum

------- [2, 5, 7, 9, 10, 11, 14, 18] sum

------- [3, 5, 6, 7, 8, 10, 16, 19] sum

------- [3, 5, 6, 7, 8, 11, 14, 20] sum

------- [3, 5, 8, 9, 10, 13, 14, 16] sum

------- [3, 6, 7, 8, 11, 13, 14, 16] sum

13_SUBTOTAL = 342

[5, 6, 17, 18, 21, 22, 23, 24] sum

------- [1, 2, 3, 4, 7, 14, 15, 20] sum

------- [1, 2, 3, 4, 12, 13, 14, 19] sum

------- [1, 2, 3, 7, 9, 10, 16, 20] sum

------- [1, 2, 3, 8, 11, 12, 14, 19] sum

------- [1, 2, 3, 11, 12, 13, 14, 16] sum

------- [1, 2, 4, 7, 10, 12, 15, 19] sum

------- [1, 2, 4, 9, 11, 14, 15, 16] sum

------- [1, 2, 7, 8, 9, 12, 14, 19] sum

------- [1, 2, 7, 9, 10, 11, 12, 20] sum

------- [1, 2, 7, 9, 12, 13, 14, 16] sum

------- [1, 2, 7, 10, 11, 12, 15, 16] sum

------- [1, 2, 8, 9, 10, 13, 15, 16] sum

------- [1, 3, 4, 7, 8, 12, 16, 19] sum

------- [1, 3, 7, 8, 9, 10, 14, 20] sum

------- [1, 3, 7, 8, 10, 14, 15, 16] sum

------- [1, 4, 7, 10, 12, 13, 14, 15] sum

------- [1, 7, 8, 10, 11, 12, 14, 15] sum

------- [2, 3, 4, 7, 11, 12, 14, 19] sum

------- [2, 3, 4, 8, 9, 13, 14, 19] sum

------- [2, 3, 4, 8, 10, 11, 15, 19] sum

------- [2, 3, 4, 9, 10, 11, 13, 20] sum

------- [2, 3, 4, 10, 11, 13, 15, 16] sum

------- [2, 3, 7, 8, 10, 12, 13, 19] sum

------- [2, 3, 8, 9, 11, 13, 14, 16] sum

------- [2, 4, 7, 8, 9, 10, 15, 19] sum

------- [2, 4, 7, 9, 10, 13, 15, 16] sum

------- [2, 7, 8, 9, 10, 11, 15, 16] sum

------- [3, 4, 8, 10, 11, 13, 14, 15] sum

------- [4, 7, 8, 9, 10, 13, 14, 15] sum

------- [4, 7, 8, 9, 11, 12, 13, 16] sum

14_SUBTOTAL = 372

[5, 10, 15, 18, 21, 22, 23, 24] sum

------- [1, 2, 3, 4, 12, 13, 14, 19] sum

------- [1, 2, 3, 6, 8, 13, 16, 19] sum

------- [1, 2, 3, 8, 9, 14, 16, 17] sum

------- [1, 2, 3, 8, 11, 12, 14, 19] sum

------- [1, 2, 3, 11, 12, 13, 14, 16] sum

------- [1, 2, 4, 6, 7, 12, 17, 19] sum

------- [1, 2, 4, 9, 12, 13, 14, 17] sum

------- [1, 2, 6, 7, 11, 12, 16, 17] sum

------- [1, 2, 6, 8, 9, 13, 16, 17] sum

------- [1, 2, 6, 8, 11, 12, 13, 19] sum

------- [1, 2, 7, 8, 9, 12, 14, 19] sum

------- [1, 2, 7, 9, 12, 13, 14, 16] sum

------- [1, 2, 8, 9, 11, 12, 14, 17] sum

------- [1, 3, 4, 7, 8, 12, 16, 19] sum

------- [1, 3, 4, 8, 11, 12, 16, 17] sum

------- [1, 3, 6, 7, 8, 14, 16, 17] sum

------- [1, 4, 6, 7, 9, 11, 14, 20] sum

------- [1, 4, 6, 7, 12, 13, 14, 17] sum

------- [1, 4, 7, 8, 9, 12, 16, 17] sum

------- [1, 6, 7, 8, 11, 12, 14, 17] sum

------- [2, 3, 4, 6, 7, 13, 16, 19] sum

------- [2, 3, 4, 6, 8, 11, 17, 19] sum

------- [2, 3, 4, 6, 11, 13, 16, 17] sum

------- [2, 3, 4, 7, 9, 14, 16, 17] sum

------- [2, 3, 4, 7, 11, 12, 14, 19] sum

------- [2, 3, 4, 8, 9, 13, 14, 19] sum

------- [2, 3, 6, 7, 8, 11, 16, 19] sum

------- [2, 3, 8, 9, 11, 13, 14, 16] sum

------- [2, 4, 6, 7, 8, 9, 17, 19] sum

------- [2, 4, 6, 7, 9, 13, 16, 17] sum

------- [2, 4, 6, 7, 11, 12, 13, 19] sum

------- [2, 4, 7, 9, 11, 12, 14, 17] sum

------- [2, 6, 7, 8, 9, 11, 16, 17] sum

------- [3, 4, 6, 7, 8, 13, 14, 19] sum

------- [3, 4, 6, 8, 11, 13, 14, 17] sum

------- [3, 6, 7, 8, 11, 13, 14, 16] sum

------- [4, 6, 7, 8, 9, 13, 14, 17] sum

------- [4, 7, 8, 9, 11, 12, 13, 16] sum

15_SUBTOTAL = 410

[6, 9, 14, 19, 21, 22, 23, 24] sum

------- [1, 2, 3, 4, 5, 11, 18, 20] sum

------- [1, 2, 3, 4, 10, 15, 16, 17] sum

------- [1, 2, 3, 4, 11, 13, 16, 18] sum

------- [1, 2, 4, 5, 7, 15, 16, 18] sum

------- [1, 2, 4, 10, 11, 12, 15, 17] sum

------- [1, 2, 5, 7, 8, 12, 17, 18] sum

------- [1, 2, 7, 10, 11, 12, 15, 16] sum

------- [1, 3, 4, 5, 7, 12, 16, 20] sum

------- [1, 3, 4, 5, 10, 13, 16, 18] sum

------- [1, 3, 4, 8, 11, 12, 16, 17] sum

------- [1, 3, 5, 8, 10, 11, 16, 18] sum

------- [1, 4, 5, 8, 12, 13, 15, 16] sum

------- [1, 4, 5, 10, 11, 12, 13, 18] sum

------- [1, 4, 7, 8, 10, 11, 15, 18] sum

------- [2, 3, 4, 5, 8, 13, 17, 18] sum

------- [2, 3, 4, 5, 10, 11, 15, 20] sum

------- [2, 3, 4, 10, 11, 13, 15, 16] sum

------- [2, 3, 5, 7, 8, 13, 16, 18] sum

------- [2, 3, 5, 7, 10, 12, 13, 20] sum

------- [2, 3, 8, 10, 11, 12, 13, 17] sum

------- [2, 5, 7, 8, 10, 12, 15, 17] sum

------- [2, 5, 7, 8, 11, 12, 13, 18] sum

------- [3, 4, 5, 7, 10, 11, 16, 18] sum

------- [4, 5, 7, 8, 11, 12, 15, 16] sum

16_SUBTOTAL = 434

[6, 13, 15, 18, 19, 22, 23, 24] sum

------- [1, 2, 3, 4, 9, 10, 17, 20] sum

------- [1, 2, 3, 5, 8, 10, 16, 21] sum

------- [1, 2, 3, 5, 11, 12, 14, 20] sum

------- [1, 2, 3, 7, 9, 10, 16, 20] sum

------- [1, 2, 3, 8, 9, 14, 16, 17] sum

------- [1, 2, 5, 7, 9, 12, 14, 20] sum

------- [1, 2, 5, 8, 10, 11, 12, 21] sum

------- [1, 2, 5, 9, 10, 12, 16, 17] sum

------- [1, 2, 7, 9, 10, 11, 12, 20] sum

------- [1, 2, 8, 9, 11, 12, 14, 17] sum

------- [1, 3, 4, 5, 7, 12, 16, 20] sum

------- [1, 3, 4, 8, 11, 12, 16, 17] sum

------- [1, 3, 7, 8, 9, 10, 14, 20] sum

------- [1, 4, 7, 8, 9, 12, 16, 17] sum

------- [1, 5, 8, 9, 10, 12, 14, 17] sum

------- [2, 3, 4, 5, 7, 10, 16, 21] sum

------- [2, 3, 4, 7, 8, 11, 14, 21] sum

------- [2, 3, 4, 7, 9, 14, 16, 17] sum

------- [2, 3, 5, 8, 9, 11, 14, 20] sum

------- [2, 4, 5, 7, 10, 11, 12, 21] sum

------- [2, 4, 7, 9, 11, 12, 14, 17] sum

------- [3, 4, 5, 7, 8, 9, 16, 20] sum

------- [3, 4, 5, 7, 8, 10, 14, 21] sum

------- [3, 4, 5, 10, 11, 12, 14, 17] sum

------- [3, 5, 7, 10, 11, 12, 14, 16] sum

------- [4, 5, 7, 8, 9, 11, 12, 20] sum

------- [4, 5, 7, 9, 10, 12, 14, 17] sum

17_SUBTOTAL = 461

[7, 9, 12, 20, 21, 22, 23, 24] sum

------- [1, 2, 3, 4, 8, 11, 18, 19] sum

------- [1, 2, 3, 4, 10, 15, 16, 17] sum

------- [1, 2, 3, 4, 11, 13, 16, 18] sum

------- [1, 2, 3, 6, 8, 13, 16, 19] sum

------- [1, 2, 5, 6, 8, 15, 16, 17] sum

------- [1, 2, 5, 6, 10, 11, 17, 18] sum

------- [1, 3, 4, 5, 6, 14, 16, 19] sum

------- [1, 3, 4, 5, 8, 10, 18, 19] sum

------- [1, 3, 4, 5, 10, 13, 16, 18] sum

------- [1, 3, 4, 6, 10, 11, 16, 19] sum

------- [1, 3, 4, 8, 10, 14, 15, 17] sum

------- [1, 3, 4, 8, 11, 13, 14, 18] sum

------- [1, 3, 5, 8, 10, 11, 16, 18] sum

------- [1, 4, 6, 8, 10, 13, 15, 17] sum

------- [2, 3, 4, 5, 8, 13, 17, 18] sum

------- [2, 3, 4, 5, 8, 14, 15, 19] sum

------- [2, 3, 4, 5, 13, 14, 15, 16] sum

------- [2, 3, 4, 6, 8, 11, 17, 19] sum

------- [2, 3, 4, 6, 11, 13, 16, 17] sum

------- [2, 3, 4, 8, 10, 11, 15, 19] sum

------- [2, 3, 4, 10, 11, 13, 15, 16] sum

------- [2, 3, 5, 6, 10, 13, 14, 19] sum

------- [2, 3, 5, 8, 11, 14, 15, 16] sum

------- [2, 4, 5, 6, 8, 13, 15, 19] sum

------- [2, 5, 6, 8, 11, 13, 15, 16] sum

------- [3, 4, 5, 6, 8, 10, 17, 19] sum

------- [3, 4, 5, 6, 10, 13, 16, 17] sum

------- [3, 4, 6, 8, 11, 13, 14, 17] sum

------- [3, 4, 8, 10, 11, 13, 14, 15] sum

------- [3, 5, 6, 8, 10, 11, 16, 17] sum

18_SUBTOTAL = 491

[7, 11, 16, 17, 20, 22, 23, 24] sum

------- [1, 2, 3, 4, 8, 13, 14, 21] sum

------- [1, 2, 3, 4, 12, 13, 14, 19] sum

------- [1, 2, 4, 5, 10, 12, 13, 21] sum

------- [1, 2, 5, 9, 10, 13, 14, 18] sum

------- [1, 3, 4, 5, 8, 10, 18, 19] sum

------- [1, 3, 4, 9, 10, 12, 15, 18] sum

------- [1, 3, 5, 8, 10, 12, 14, 19] sum

------- [1, 5, 6, 8, 9, 12, 15, 18] sum

------- [1, 5, 6, 8, 10, 12, 13, 19] sum

------- [2, 3, 4, 5, 6, 12, 15, 21] sum

------- [2, 3, 4, 5, 8, 14, 15, 19] sum

------- [2, 3, 4, 6, 9, 12, 13, 21] sum

------- [2, 3, 4, 8, 9, 13, 14, 19] sum

------- [2, 3, 5, 6, 9, 14, 15, 18] sum

------- [2, 3, 5, 6, 10, 13, 14, 19] sum

------- [2, 4, 5, 6, 8, 13, 15, 19] sum

------- [2, 4, 5, 8, 9, 10, 13, 21] sum

------- [2, 4, 5, 9, 10, 12, 13, 19] sum

------- [3, 5, 6, 8, 9, 10, 12, 21] sum

19_SUBTOTAL = 510

[7, 12, 15, 16, 21, 22, 23, 24] sum

------- [1, 2, 3, 4, 5, 11, 18, 20] sum

------- [1, 2, 3, 4, 8, 11, 18, 19] sum

------- [1, 2, 3, 4, 9, 10, 17, 20] sum

------- [1, 2, 3, 5, 6, 8, 19, 20] sum

------- [1, 2, 4, 8, 9, 11, 17, 18] sum

------- [1, 2, 5, 6, 8, 9, 17, 20] sum

------- [1, 2, 5, 6, 10, 11, 17, 18] sum

------- [1, 2, 5, 9, 10, 13, 14, 18] sum

------- [1, 2, 6, 9, 10, 11, 14, 19] sum

------- [1, 3, 4, 5, 8, 10, 18, 19] sum

------- [1, 3, 4, 8, 11, 13, 14, 18] sum

------- [1, 3, 5, 6, 8, 13, 14, 20] sum

------- [1, 3, 6, 8, 10, 11, 13, 20] sum

------- [1, 4, 5, 8, 9, 10, 17, 18] sum

------- [2, 3, 4, 5, 6, 11, 17, 20] sum

------- [2, 3, 4, 5, 8, 13, 17, 18] sum

------- [2, 3, 4, 5, 9, 13, 14, 20] sum

------- [2, 3, 4, 6, 8, 11, 17, 19] sum

------- [2, 3, 4, 8, 9, 13, 14, 19] sum

------- [2, 3, 4, 9, 10, 11, 13, 20] sum

------- [2, 3, 5, 6, 10, 13, 14, 19] sum

------- [2, 3, 5, 8, 9, 11, 14, 20] sum

------- [2, 5, 6, 8, 9, 11, 13, 20] sum

------- [2, 5, 6, 9, 10, 13, 14, 17] sum

------- [3, 4, 5, 6, 8, 10, 17, 19] sum

------- [3, 4, 6, 8, 11, 13, 14, 17] sum

------- [4, 5, 6, 8, 9, 11, 14, 19] sum

------- [4, 5, 8, 9, 10, 11, 13, 18] sum

20_SUBTOTAL = 538

[8, 9, 17, 18, 20, 21, 23, 24] sum

------- [1, 2, 3, 4, 12, 13, 14, 19] sum

------- [1, 2, 3, 5, 10, 12, 16, 19] sum

------- [1, 2, 3, 6, 7, 11, 14, 22] sum

------- [1, 2, 3, 11, 12, 13, 14, 16] sum

------- [1, 2, 4, 7, 10, 12, 15, 19] sum

------- [1, 2, 5, 7, 12, 14, 15, 16] sum

------- [1, 2, 7, 10, 11, 12, 15, 16] sum

------- [1, 3, 4, 5, 6, 14, 16, 19] sum

------- [1, 3, 4, 5, 10, 11, 12, 22] sum

------- [1, 3, 4, 6, 10, 11, 16, 19] sum

------- [1, 3, 5, 6, 7, 10, 14, 22] sum

------- [1, 3, 5, 10, 12, 13, 14, 16] sum

------- [1, 4, 5, 6, 11, 12, 14, 19] sum

------- [1, 4, 6, 7, 11, 14, 15, 16] sum

------- [1, 4, 7, 10, 12, 13, 14, 15] sum

------- [2, 3, 4, 5, 7, 12, 13, 22] sum

------- [2, 3, 4, 5, 13, 14, 15, 16] sum

------- [2, 3, 4, 6, 7, 13, 16, 19] sum

------- [2, 3, 4, 7, 11, 12, 14, 19] sum

------- [2, 3, 4, 10, 11, 13, 15, 16] sum

------- [2, 3, 5, 6, 10, 13, 14, 19] sum

------- [2, 4, 5, 11, 12, 13, 14, 15] sum

------- [2, 4, 6, 7, 11, 12, 13, 19] sum

------- [2, 6, 7, 10, 11, 13, 14, 15] sum

------- [3, 4, 5, 7, 10, 12, 14, 19] sum

------- [3, 5, 7, 10, 11, 12, 14, 16] sum

------- [4, 5, 6, 7, 10, 12, 13, 19] sum

------- [5, 6, 7, 10, 11, 12, 13, 16] sum

21_SUBTOTAL = 566

[8, 11, 13, 19, 20, 22, 23, 24] sum

------- [1, 2, 3, 4, 10, 15, 16, 17] sum

------- [1, 2, 4, 5, 7, 15, 16, 18] sum

------- [1, 2, 4, 5, 12, 14, 15, 17] sum

------- [1, 2, 5, 7, 12, 14, 15, 16] sum

------- [1, 2, 5, 9, 10, 12, 16, 17] sum

------- [1, 2, 6, 7, 10, 14, 15, 17] sum

------- [1, 3, 4, 6, 9, 12, 17, 18] sum

------- [1, 3, 4, 9, 10, 12, 15, 18] sum

------- [1, 3, 6, 7, 9, 12, 16, 18] sum

------- [1, 4, 5, 6, 9, 14, 16, 17] sum

------- [1, 4, 5, 9, 10, 14, 15, 16] sum

------- [1, 5, 6, 7, 10, 12, 16, 17] sum

------- [2, 3, 4, 5, 6, 12, 15, 21] sum

------- [2, 3, 4, 5, 7, 10, 16, 21] sum

------- [2, 3, 4, 7, 9, 14, 16, 17] sum

------- [2, 3, 5, 6, 9, 14, 15, 18] sum

------- [2, 4, 5, 6, 7, 15, 16, 17] sum

------- [3, 4, 6, 9, 10, 12, 15, 17] sum

------- [3, 6, 7, 9, 10, 12, 15, 16] sum

------- [4, 5, 6, 7, 9, 12, 15, 18] sum

------- [4, 5, 7, 9, 10, 12, 14, 17] sum

22_SUBTOTAL = 587

[8, 13, 16, 19, 20, 21, 22, 23] sum

------- [1, 2, 3, 6, 7, 9, 12, 24] sum

------- [1, 2, 4, 5, 12, 14, 15, 17] sum

------- [1, 2, 4, 10, 11, 12, 15, 17] sum

------- [1, 2, 5, 6, 10, 11, 17, 18] sum

------- [1, 2, 6, 7, 10, 14, 15, 17] sum

------- [1, 3, 4, 6, 9, 12, 17, 18] sum

------- [1, 3, 4, 9, 10, 12, 15, 18] sum

------- [2, 3, 4, 5, 7, 10, 11, 24] sum

------- [2, 3, 5, 6, 9, 14, 15, 18] sum

------- [2, 3, 6, 9, 10, 11, 15, 18] sum

------- [2, 4, 7, 9, 11, 12, 14, 17] sum

------- [2, 5, 7, 9, 10, 11, 14, 18] sum

------- [3, 4, 5, 6, 11, 12, 15, 18] sum

------- [3, 4, 5, 10, 11, 12, 14, 17] sum

------- [3, 4, 6, 9, 10, 12, 15, 17] sum

------- [4, 5, 6, 7, 9, 12, 15, 18] sum

------- [4, 5, 7, 9, 10, 12, 14, 17] sum

23_SUBTOTAL = 604

[8, 15, 16, 17, 18, 21, 23, 24] sum

------- [1, 2, 3, 4, 12, 13, 14, 19] sum

------- [1, 2, 3, 5, 9, 10, 14, 22] sum

------- [1, 2, 3, 5, 11, 12, 14, 20] sum

------- [1, 2, 3, 6, 7, 11, 14, 22] sum

------- [1, 2, 4, 7, 9, 11, 12, 22] sum

------- [1, 2, 5, 6, 9, 10, 13, 22] sum

------- [1, 2, 5, 6, 11, 12, 13, 20] sum

------- [1, 2, 5, 7, 9, 12, 14, 20] sum

------- [1, 2, 6, 9, 10, 11, 14, 19] sum

------- [1, 2, 7, 9, 10, 11, 12, 20] sum

------- [1, 3, 4, 5, 10, 11, 12, 22] sum

------- [1, 3, 5, 6, 7, 10, 14, 22] sum

------- [1, 4, 5, 6, 11, 12, 14, 19] sum

------- [1, 4, 5, 7, 9, 10, 12, 22] sum

------- [1, 4, 6, 7, 9, 11, 14, 20] sum

------- [2, 3, 4, 5, 6, 7, 19, 20] sum

------- [2, 3, 4, 5, 7, 12, 13, 22] sum

------- [2, 3, 4, 5, 9, 13, 14, 20] sum

------- [2, 3, 4, 7, 11, 12, 14, 19] sum

------- [2, 3, 4, 9, 10, 11, 13, 20] sum

------- [2, 3, 5, 6, 10, 13, 14, 19] sum

------- [2, 3, 5, 7, 10, 12, 13, 20] sum

------- [2, 4, 5, 9, 10, 12, 13, 19] sum

------- [2, 4, 6, 7, 11, 12, 13, 19] sum

------- [2, 5, 6, 7, 9, 10, 11, 22] sum

------- [3, 4, 5, 6, 7, 13, 14, 20] sum

------- [3, 4, 5, 7, 10, 12, 14, 19] sum

------- [3, 4, 6, 7, 10, 11, 13, 20] sum

------- [4, 5, 6, 7, 10, 12, 13, 19] sum

24_SUBTOTAL = 633

[9, 10, 13, 18, 21, 22, 23, 24] sum

------- [1, 2, 3, 4, 7, 14, 15, 20] sum

------- [1, 2, 3, 5, 6, 8, 19, 20] sum

------- [1, 2, 3, 5, 11, 12, 14, 20] sum

------- [1, 2, 3, 8, 11, 12, 14, 19] sum

------- [1, 2, 4, 5, 12, 14, 15, 17] sum

------- [1, 2, 4, 6, 7, 12, 17, 19] sum

------- [1, 2, 5, 6, 8, 15, 16, 17] sum

------- [1, 2, 5, 7, 12, 14, 15, 16] sum

------- [1, 2, 6, 7, 8, 11, 15, 20] sum

------- [1, 2, 6, 7, 11, 12, 16, 17] sum

------- [1, 3, 4, 5, 6, 14, 16, 19] sum

------- [1, 3, 4, 5, 7, 12, 16, 20] sum

------- [1, 3, 4, 7, 8, 12, 16, 19] sum

------- [1, 3, 4, 8, 11, 12, 16, 17] sum

------- [1, 3, 6, 7, 8, 14, 16, 17] sum

------- [1, 4, 5, 6, 11, 12, 14, 19] sum

------- [1, 4, 6, 7, 11, 14, 15, 16] sum

------- [1, 6, 7, 8, 11, 12, 14, 17] sum

------- [2, 3, 4, 5, 6, 7, 19, 20] sum

------- [2, 3, 4, 5, 6, 11, 17, 20] sum

------- [2, 3, 4, 5, 8, 14, 15, 19] sum

------- [2, 3, 4, 6, 8, 11, 17, 19] sum

------- [2, 3, 4, 7, 11, 12, 14, 19] sum

------- [2, 3, 5, 6, 7, 11, 16, 20] sum

------- [2, 3, 5, 8, 11, 14, 15, 16] sum

------- [2, 3, 6, 7, 8, 11, 16, 19] sum

------- [2, 4, 5, 6, 7, 15, 16, 17] sum

------- [3, 5, 6, 7, 8, 11, 14, 20] sum

------- [4, 5, 6, 7, 8, 14, 15, 17] sum

------- [4, 5, 7, 8, 11, 12, 15, 16] sum

25_SUBTOTAL = 663

[9, 12, 17, 18, 19, 20, 23, 24] sum

------- [1, 2, 3, 4, 8, 13, 14, 21] sum

------- [1, 2, 3, 5, 8, 10, 16, 21] sum

------- [1, 2, 3, 6, 7, 11, 14, 22] sum

------- [1, 2, 4, 5, 6, 11, 16, 21] sum

------- [1, 2, 4, 7, 8, 10, 15, 21] sum

------- [1, 3, 4, 6, 8, 11, 13, 22] sum

------- [1, 3, 5, 6, 7, 10, 14, 22] sum

------- [1, 3, 7, 8, 10, 14, 15, 16] sum

------- [1, 4, 5, 6, 7, 8, 15, 22] sum

------- [1, 4, 5, 6, 8, 11, 14, 21] sum

------- [1, 4, 6, 7, 11, 14, 15, 16] sum

------- [1, 6, 7, 8, 10, 13, 15, 16] sum

------- [2, 3, 4, 5, 7, 10, 16, 21] sum

------- [2, 3, 4, 5, 13, 14, 15, 16] sum

------- [2, 3, 4, 7, 8, 11, 14, 21] sum

------- [2, 3, 4, 10, 11, 13, 15, 16] sum

------- [2, 3, 5, 8, 11, 14, 15, 16] sum

------- [2, 4, 6, 7, 8, 11, 13, 21] sum

------- [2, 5, 6, 8, 11, 13, 15, 16] sum

------- [2, 6, 7, 10, 11, 13, 14, 15] sum

------- [3, 4, 5, 7, 8, 10, 14, 21] sum

------- [3, 4, 8, 10, 11, 13, 14, 15] sum

------- [3, 6, 7, 8, 11, 13, 14, 16] sum

------- [4, 5, 6, 7, 8, 10, 13, 21] sum

26_SUBTOTAL = 687

[9, 14, 15, 16, 20, 21, 23, 24] sum

------- [1, 2, 3, 4, 8, 11, 18, 19] sum

------- [1, 2, 3, 5, 8, 12, 13, 22] sum

------- [1, 2, 4, 6, 7, 12, 17, 19] sum

------- [1, 2, 5, 6, 7, 10, 18, 19] sum

------- [1, 2, 5, 6, 10, 11, 17, 18] sum

------- [1, 2, 5, 7, 8, 12, 17, 18] sum

------- [1, 2, 6, 8, 11, 12, 13, 19] sum

------- [1, 3, 4, 5, 8, 10, 18, 19] sum

------- [1, 3, 4, 5, 10, 11, 12, 22] sum

------- [1, 3, 4, 6, 8, 11, 13, 22] sum

------- [1, 4, 5, 10, 11, 12, 13, 18] sum

------- [1, 4, 6, 7, 8, 11, 17, 18] sum

------- [1, 5, 6, 8, 10, 12, 13, 19] sum

------- [2, 3, 4, 5, 7, 12, 13, 22] sum

------- [2, 3, 4, 5, 8, 13, 17, 18] sum

------- [2, 3, 4, 6, 8, 11, 17, 19] sum

------- [2, 3, 5, 7, 8, 11, 12, 22] sum

------- [2, 3, 7, 8, 10, 12, 13, 19] sum

------- [2, 3, 8, 10, 11, 12, 13, 17] sum

------- [2, 4, 6, 7, 11, 12, 13, 19] sum

------- [2, 5, 7, 8, 11, 12, 13, 18] sum

------- [3, 4, 5, 6, 8, 10, 17, 19] sum

------- [4, 5, 6, 7, 10, 12, 13, 19] sum

------- [4, 5, 6, 10, 11, 12, 13, 17] sum

------- [5, 6, 7, 8, 10, 11, 12, 19] sum

27_SUBTOTAL = 712

[9, 14, 15, 17, 18, 22, 23, 24] sum

------- [1, 2, 3, 5, 6, 8, 19, 20] sum

------- [1, 2, 3, 5, 6, 13, 16, 20] sum

------- [1, 2, 3, 5, 8, 10, 16, 21] sum

------- [1, 2, 3, 5, 10, 12, 16, 19] sum

------- [1, 2, 3, 6, 8, 13, 16, 19] sum

------- [1, 2, 4, 5, 6, 11, 16, 21] sum

------- [1, 2, 4, 5, 10, 12, 13, 21] sum

------- [1, 2, 5, 6, 11, 12, 13, 20] sum

------- [1, 2, 5, 8, 10, 11, 12, 21] sum

------- [1, 2, 6, 8, 11, 12, 13, 19] sum

------- [1, 3, 4, 5, 7, 12, 16, 20] sum

------- [1, 3, 4, 6, 10, 11, 16, 19] sum

------- [1, 3, 4, 7, 8, 12, 16, 19] sum

------- [1, 3, 6, 8, 10, 11, 13, 20] sum

------- [1, 5, 6, 8, 10, 12, 13, 19] sum

------- [2, 3, 4, 5, 6, 7, 19, 20] sum

------- [2, 3, 4, 5, 7, 10, 16, 21] sum

------- [2, 3, 4, 6, 7, 13, 16, 19] sum

------- [2, 3, 5, 6, 7, 11, 16, 20] sum

------- [2, 3, 5, 7, 10, 12, 13, 20] sum

------- [2, 3, 6, 7, 8, 11, 16, 19] sum

------- [2, 3, 7, 8, 10, 12, 13, 19] sum

------- [2, 4, 5, 7, 10, 11, 12, 21] sum

------- [2, 4, 6, 7, 8, 11, 13, 21] sum

------- [2, 4, 6, 7, 11, 12, 13, 19] sum

------- [3, 4, 6, 7, 10, 11, 13, 20] sum

------- [3, 5, 6, 7, 8, 10, 16, 19] sum

------- [4, 5, 6, 7, 8, 10, 13, 21] sum

------- [4, 5, 6, 7, 10, 12, 13, 19] sum

------- [5, 6, 7, 8, 10, 11, 12, 19] sum

------- [5, 6, 7, 10, 11, 12, 13, 16] sum

28_SUBTOTAL = 743

[10, 12, 15, 17, 20, 21, 23, 24] sum

------- [1, 2, 3, 4, 7, 9, 16, 22] sum

------- [1, 2, 3, 4, 8, 11, 18, 19] sum

------- [1, 2, 3, 4, 11, 13, 16, 18] sum

------- [1, 2, 3, 6, 7, 11, 14, 22] sum

------- [1, 2, 3, 6, 8, 13, 16, 19] sum

------- [1, 2, 4, 7, 8, 9, 18, 19] sum

------- [1, 2, 4, 7, 9, 13, 16, 18] sum

------- [1, 2, 6, 7, 11, 13, 14, 18] sum

------- [1, 2, 7, 8, 9, 11, 16, 18] sum

------- [1, 3, 4, 5, 6, 14, 16, 19] sum

------- [1, 3, 4, 6, 8, 11, 13, 22] sum

------- [1, 3, 4, 7, 8, 9, 14, 22] sum

------- [1, 3, 4, 8, 11, 13, 14, 18] sum

------- [1, 4, 6, 7, 8, 9, 13, 22] sum

------- [1, 4, 7, 8, 9, 13, 14, 18] sum

------- [2, 3, 4, 6, 7, 13, 16, 19] sum

------- [2, 3, 4, 8, 9, 13, 14, 19] sum

------- [2, 3, 5, 7, 8, 13, 16, 18] sum

------- [2, 3, 6, 7, 8, 11, 16, 19] sum

------- [2, 3, 8, 9, 11, 13, 14, 16] sum

------- [2, 4, 5, 6, 9, 11, 16, 19] sum

------- [3, 4, 6, 7, 8, 13, 14, 19] sum

------- [3, 6, 7, 8, 11, 13, 14, 16] sum

------- [4, 5, 6, 8, 9, 11, 14, 19] sum

------- [4, 5, 6, 9, 11, 13, 14, 16] sum

29_SUBTOTAL = 768

[10, 13, 14, 17, 19, 22, 23, 24] sum

------- [1, 2, 3, 4, 5, 11, 18, 20] sum

------- [1, 2, 3, 6, 9, 12, 15, 20] sum

------- [1, 2, 4, 5, 6, 11, 16, 21] sum

------- [1, 2, 4, 5, 7, 9, 18, 20] sum

------- [1, 2, 4, 5, 7, 15, 16, 18] sum

------- [1, 2, 6, 7, 8, 11, 15, 20] sum

------- [1, 2, 7, 8, 9, 11, 16, 18] sum

------- [1, 3, 4, 5, 7, 12, 16, 20] sum

------- [1, 3, 6, 7, 9, 12, 16, 18] sum

------- [1, 5, 6, 8, 9, 12, 15, 18] sum

------- [2, 3, 4, 5, 6, 12, 15, 21] sum

------- [2, 3, 5, 6, 7, 11, 16, 20] sum

------- [2, 3, 6, 8, 9, 11, 12, 21] sum

------- [2, 3, 7, 8, 9, 12, 15, 18] sum

------- [3, 4, 5, 6, 11, 12, 15, 18] sum

------- [3, 4, 5, 7, 8, 9, 16, 20] sum

------- [4, 5, 6, 7, 9, 12, 15, 18] sum

------- [4, 5, 7, 8, 9, 11, 12, 20] sum

------- [4, 5, 7, 8, 11, 12, 15, 16] sum

30_SUBTOTAL = 787

[11, 12, 13, 18, 20, 21, 23, 24] sum

------- [1, 2, 3, 4, 7, 9, 16, 22] sum

------- [1, 2, 3, 4, 10, 15, 16, 17] sum

------- [1, 2, 3, 5, 9, 10, 14, 22] sum

------- [1, 2, 3, 8, 9, 14, 16, 17] sum

------- [1, 2, 4, 5, 8, 9, 15, 22] sum

------- [1, 2, 5, 6, 8, 15, 16, 17] sum

------- [1, 2, 6, 7, 10, 14, 15, 17] sum

------- [1, 3, 4, 5, 6, 14, 16, 19] sum

------- [1, 3, 4, 7, 8, 9, 14, 22] sum

------- [1, 3, 4, 8, 10, 14, 15, 17] sum

------- [1, 3, 5, 6, 7, 10, 14, 22] sum

------- [1, 3, 6, 7, 8, 14, 16, 17] sum

------- [1, 3, 7, 8, 10, 14, 15, 16] sum

------- [1, 4, 5, 6, 7, 8, 15, 22] sum

------- [1, 4, 5, 6, 9, 14, 16, 17] sum

------- [1, 4, 5, 9, 10, 14, 15, 16] sum

------- [1, 4, 6, 7, 9, 10, 16, 19] sum

------- [2, 3, 4, 5, 8, 14, 15, 19] sum

------- [2, 3, 4, 7, 9, 14, 16, 17] sum

------- [2, 3, 5, 8, 9, 10, 16, 19] sum

------- [2, 4, 5, 6, 7, 15, 16, 17] sum

------- [2, 4, 5, 8, 9, 14, 15, 17] sum

------- [2, 4, 6, 7, 8, 9, 17, 19] sum

------- [2, 4, 7, 8, 9, 10, 15, 19] sum

------- [2, 5, 7, 8, 9, 14, 15, 16] sum

------- [3, 4, 5, 6, 8, 10, 17, 19] sum

------- [3, 5, 6, 7, 8, 10, 16, 19] sum

------- [4, 5, 6, 7, 8, 14, 15, 17] sum

------- [5, 6, 7, 8, 9, 10, 16, 17] sum

31_SUBTOTAL = 816

[11, 14, 17, 18, 19, 20, 22, 23] sum

------- [1, 2, 3, 4, 5, 10, 13, 24] sum

------- [1, 2, 3, 5, 8, 10, 16, 21] sum

------- [1, 2, 3, 6, 7, 9, 12, 24] sum

------- [1, 2, 4, 5, 10, 12, 13, 21] sum

------- [1, 2, 4, 7, 8, 10, 15, 21] sum

------- [1, 2, 5, 7, 8, 9, 10, 24] sum

------- [1, 2, 8, 9, 10, 13, 15, 16] sum

------- [1, 4, 5, 8, 12, 13, 15, 16] sum

------- [1, 6, 7, 8, 10, 13, 15, 16] sum

------- [2, 3, 4, 5, 6, 12, 15, 21] sum

------- [2, 3, 4, 5, 7, 10, 16, 21] sum

------- [2, 3, 4, 6, 9, 12, 13, 21] sum

------- [2, 4, 5, 8, 9, 10, 13, 21] sum

------- [2, 4, 7, 9, 10, 13, 15, 16] sum

------- [3, 5, 6, 8, 9, 10, 12, 21] sum

------- [3, 6, 7, 9, 10, 12, 15, 16] sum

------- [4, 5, 6, 7, 8, 10, 13, 21] sum

32_SUBTOTAL = 833

[11, 15, 16, 18, 19, 20, 21, 24] sum

------- [1, 2, 3, 4, 8, 9, 14, 23] sum

------- [1, 2, 3, 5, 6, 10, 14, 23] sum

------- [1, 2, 3, 5, 8, 12, 13, 22] sum

------- [1, 2, 3, 5, 9, 10, 14, 22] sum

------- [1, 2, 3, 7, 8, 10, 12, 23] sum

------- [1, 2, 4, 5, 9, 10, 12, 23] sum

------- [1, 2, 4, 6, 8, 9, 13, 23] sum

------- [1, 2, 4, 9, 12, 13, 14, 17] sum

------- [1, 2, 5, 6, 9, 10, 13, 22] sum

------- [1, 3, 4, 6, 7, 8, 14, 23] sum

------- [1, 3, 4, 7, 8, 9, 14, 22] sum

------- [1, 3, 5, 6, 7, 10, 14, 22] sum

------- [1, 4, 5, 6, 7, 10, 12, 23] sum

------- [1, 4, 5, 7, 9, 10, 12, 22] sum

------- [1, 4, 6, 7, 8, 9, 13, 22] sum

------- [1, 4, 6, 7, 12, 13, 14, 17] sum

------- [1, 5, 8, 9, 10, 12, 14, 17] sum

------- [2, 3, 4, 5, 7, 12, 13, 22] sum

------- [2, 3, 5, 8, 12, 13, 14, 17] sum

------- [2, 5, 6, 9, 10, 13, 14, 17] sum

------- [2, 7, 8, 9, 10, 12, 13, 17] sum

------- [4, 5, 6, 7, 8, 9, 10, 23] sum

------- [4, 5, 7, 9, 10, 12, 14, 17] sum

------- [4, 6, 7, 8, 9, 13, 14, 17] sum

33_SUBTOTAL = 857

[12, 13, 17, 18, 19, 20, 21, 24] sum

------- [1, 2, 3, 4, 6, 7, 16, 23] sum

------- [1, 2, 3, 4, 7, 9, 16, 22] sum

------- [1, 2, 3, 4, 8, 9, 14, 23] sum

------- [1, 2, 3, 5, 6, 10, 14, 23] sum

------- [1, 2, 3, 5, 9, 10, 14, 22] sum

------- [1, 2, 3, 6, 7, 11, 14, 22] sum

------- [1, 2, 4, 5, 6, 8, 15, 23] sum

------- [1, 2, 4, 5, 8, 9, 15, 22] sum

------- [1, 2, 4, 9, 11, 14, 15, 16] sum

------- [1, 3, 4, 6, 7, 8, 14, 23] sum

------- [1, 3, 4, 7, 8, 9, 14, 22] sum

------- [1, 3, 5, 6, 7, 10, 14, 22] sum

------- [1, 3, 7, 8, 10, 14, 15, 16] sum

------- [1, 4, 5, 6, 7, 8, 15, 22] sum

------- [1, 4, 5, 9, 10, 14, 15, 16] sum

------- [1, 4, 6, 7, 11, 14, 15, 16] sum

------- [2, 3, 5, 8, 11, 14, 15, 16] sum

------- [2, 4, 6, 7, 8, 9, 11, 23] sum

------- [2, 5, 6, 7, 9, 10, 11, 22] sum

------- [2, 5, 7, 8, 9, 14, 15, 16] sum

------- [2, 7, 8, 9, 10, 11, 15, 16] sum

------- [3, 4, 5, 6, 8, 10, 11, 23] sum

------- [3, 4, 5, 8, 9, 10, 11, 22] sum

------- [4, 5, 6, 7, 8, 9, 10, 23] sum

34_SUBTOTAL = 881

[12, 14, 15, 18, 19, 21, 22, 23] sum

------- [1, 2, 3, 4, 5, 10, 13, 24] sum

------- [1, 2, 3, 4, 9, 10, 17, 20] sum

------- [1, 2, 3, 5, 6, 13, 16, 20] sum

------- [1, 2, 3, 5, 8, 10, 11, 24] sum

------- [1, 2, 3, 7, 9, 10, 16, 20] sum

------- [1, 2, 5, 6, 8, 9, 17, 20] sum

------- [1, 2, 5, 7, 8, 9, 10, 24] sum

------- [1, 2, 6, 8, 9, 13, 16, 17] sum

------- [1, 3, 4, 6, 7, 10, 17, 20] sum

------- [1, 3, 6, 8, 10, 11, 13, 20] sum

------- [1, 4, 6, 9, 10, 11, 16, 17] sum

------- [1, 6, 7, 8, 9, 10, 13, 20] sum

------- [2, 3, 4, 5, 6, 11, 17, 20] sum

------- [2, 3, 4, 5, 7, 10, 11, 24] sum

------- [2, 3, 4, 6, 11, 13, 16, 17] sum

------- [2, 3, 4, 9, 10, 11, 13, 20] sum

------- [2, 3, 5, 6, 7, 11, 16, 20] sum

------- [2, 4, 5, 6, 7, 9, 17, 20] sum

------- [2, 4, 6, 7, 9, 13, 16, 17] sum

------- [2, 5, 6, 8, 9, 11, 13, 20] sum

------- [2, 6, 7, 8, 9, 11, 16, 17] sum

------- [3, 4, 5, 6, 10, 13, 16, 17] sum

------- [3, 4, 5, 7, 8, 9, 16, 20] sum

------- [3, 4, 6, 7, 10, 11, 13, 20] sum

------- [3, 5, 6, 8, 10, 11, 16, 17] sum

------- [5, 6, 7, 8, 9, 10, 16, 17] sum

35_SUBTOTAL = 907

[12, 15, 16, 17, 18, 19, 23, 24] sum

------- [1, 2, 3, 4, 8, 13, 14, 21] sum

------- [1, 2, 3, 5, 9, 10, 14, 22] sum

------- [1, 2, 3, 6, 7, 11, 14, 22] sum

------- [1, 2, 5, 6, 9, 10, 13, 22] sum

------- [1, 3, 4, 6, 8, 11, 13, 22] sum

------- [1, 3, 4, 7, 8, 9, 14, 22] sum

------- [1, 3, 5, 6, 7, 10, 14, 22] sum

------- [1, 3, 5, 6, 8, 13, 14, 20] sum

------- [1, 3, 6, 8, 10, 11, 13, 20] sum

------- [1, 3, 7, 8, 9, 10, 14, 20] sum

------- [1, 4, 5, 6, 8, 11, 14, 21] sum

------- [1, 4, 6, 7, 8, 9, 13, 22] sum

------- [1, 4, 6, 7, 9, 11, 14, 20] sum

------- [1, 6, 7, 8, 9, 10, 13, 20] sum

------- [2, 3, 4, 5, 9, 13, 14, 20] sum

------- [2, 3, 4, 7, 8, 11, 14, 21] sum

------- [2, 3, 4, 9, 10, 11, 13, 20] sum

------- [2, 3, 5, 8, 9, 11, 14, 20] sum

------- [2, 4, 5, 8, 9, 10, 13, 21] sum

------- [2, 4, 6, 7, 8, 11, 13, 21] sum

------- [2, 5, 6, 7, 9, 10, 11, 22] sum

------- [2, 5, 6, 8, 9, 11, 13, 20] sum

------- [3, 4, 5, 6, 7, 13, 14, 20] sum

------- [3, 4, 5, 7, 8, 10, 14, 21] sum

------- [3, 4, 5, 8, 9, 10, 11, 22] sum

------- [3, 4, 6, 7, 10, 11, 13, 20] sum

------- [3, 5, 6, 7, 8, 11, 14, 20] sum

------- [4, 5, 6, 7, 8, 10, 13, 21] sum

36_SUBTOTAL = 935

[13, 14, 15, 17, 18, 21, 22, 24] sum

------- [1, 2, 3, 4, 6, 7, 16, 23] sum

------- [1, 2, 3, 5, 6, 8, 19, 20] sum

------- [1, 2, 3, 5, 10, 12, 16, 19] sum

------- [1, 2, 3, 7, 8, 10, 12, 23] sum

------- [1, 2, 3, 7, 9, 10, 16, 20] sum

------- [1, 2, 4, 5, 9, 10, 12, 23] sum

------- [1, 2, 4, 6, 7, 11, 12, 23] sum

------- [1, 2, 7, 9, 10, 11, 12, 20] sum

------- [1, 3, 4, 5, 7, 12, 16, 20] sum

------- [1, 3, 4, 6, 10, 11, 16, 19] sum

------- [1, 3, 4, 7, 8, 12, 16, 19] sum

------- [1, 4, 5, 6, 7, 10, 12, 23] sum

------- [1, 4, 6, 7, 9, 10, 16, 19] sum

------- [2, 3, 4, 5, 6, 7, 19, 20] sum

------- [2, 3, 5, 6, 7, 11, 16, 20] sum

------- [2, 3, 5, 8, 9, 10, 16, 19] sum

------- [2, 3, 6, 7, 8, 11, 16, 19] sum

------- [2, 4, 5, 6, 9, 11, 16, 19] sum

------- [2, 4, 6, 7, 8, 9, 11, 23] sum

------- [2, 5, 8, 9, 10, 11, 12, 19] sum

------- [3, 4, 5, 6, 8, 10, 11, 23] sum

------- [3, 4, 5, 7, 8, 9, 16, 20] sum

------- [3, 5, 6, 7, 8, 10, 16, 19] sum

------- [4, 5, 6, 7, 8, 9, 10, 23] sum

------- [4, 5, 7, 8, 9, 11, 12, 20] sum

------- [5, 6, 7, 8, 10, 11, 12, 19] sum

37_SUBTOTAL = 961

At present there is no reason to assume that 961 = 31

(2004.03.01: More anon.)

The null vector u = [1,1,1,1,1,1,1,1,1 | 3] in the Lorentzian space

of the

n(x) = [0, x1, ..., x8 | k].

Then the inner product of u and n(x) is

<u, n(x)> = x1 + ... + x8 - 3k.

Of the 361 octet triples (x,y,z) above there are 72 that satisfy the following interesting property:

<u, n(x)> = <u, n(y)> = <u, n(z)> = -18.

Therefore, if

x1

then

x1 + ... + x8 = 72 and y1 + ... + y8 = 90 and z1 + ... + z8 = 138.

If we define

p(x) = [18, x1, ..., x8 | k], then

<u, p(x)> = <u, p(y)> = <u, p(z)> = 0.

The 72 triples are indicated in the list below:

[2, 14, 17, 19, 20, 21, 22, 23: sum 138] sum

------- [1, 3, 4, 9, 10, 12, 15, 18: sum 72] sum

------- [1, 3, 5, 8, 10, 11, 16, 18: sum 72] sum

------- [1, 3, 6, 7, 9, 12, 16, 18: sum 72] sum

1_SUBTOTAL = 3

[2, 15, 17, 18, 19, 21, 22, 24: sum 138] sum

------- [1, 3, 6, 8, 10, 11, 13, 20: sum 72] sum

------- [1, 3, 7, 8, 9, 10, 14, 20: sum 72] sum

------- [1, 4, 5, 8, 9, 12, 13, 20: sum 72] sum

------- [1, 4, 6, 7, 9, 11, 14, 20: sum 72] sum

------- [3, 4, 5, 6, 7, 13, 14, 20: sum 72] sum

------- [3, 4, 5, 7, 8, 9, 16, 20: sum 72] sum

------- [4, 5, 6, 7, 8, 9, 10, 23: sum 72] sum

2_SUBTOTAL = 10

[3, 12, 17, 19, 20, 21, 22, 24: sum 138] sum

------- [1, 2, 4, 9, 11, 14, 15, 16: sum 72] sum

------- [1, 2, 5, 9, 10, 13, 14, 18: sum 72] sum

------- [1, 2, 6, 7, 11, 13, 14, 18: sum 72] sum

------- [1, 2, 7, 8, 9, 11, 16, 18: sum 72] sum

------- [1, 4, 5, 7, 8, 14, 15, 18: sum 72] sum

------- [4, 5, 6, 7, 8, 9, 10, 23: sum 72] sum

3_SUBTOTAL = 16

[3, 13, 16, 18, 20, 21, 23, 24: sum 138] sum

------- [1, 2, 4, 10, 11, 12, 15, 17: sum 72] sum

------- [1, 2, 6, 7, 10, 14, 15, 17: sum 72] sum

------- [1, 2, 6, 9, 10, 11, 14, 19: sum 72] sum

------- [1, 2, 7, 8, 9, 12, 14, 19: sum 72] sum

------- [1, 4, 5, 6, 11, 12, 14, 19: sum 72] sum

------- [2, 4, 6, 7, 8, 9, 17, 19: sum 72] sum

------- [2, 5, 6, 7, 9, 10, 11, 22: sum 72] sum

4_SUBTOTAL = 23

[3, 14, 15, 18, 19, 22, 23, 24: sum 138] sum

------- [1, 2, 5, 9, 10, 12, 16, 17: sum 72] sum

------- [1, 2, 6, 7, 11, 12, 16, 17: sum 72] sum

------- [1, 2, 6, 8, 9, 13, 16, 17: sum 72] sum

------- [1, 2, 7, 9, 10, 11, 12, 20: sum 72] sum

------- [1, 4, 5, 8, 9, 12, 13, 20: sum 72] sum

------- [2, 4, 5, 7, 10, 11, 12, 21: sum 72] sum

------- [2, 4, 5, 8, 9, 10, 13, 21: sum 72] sum

------- [2, 4, 6, 7, 8, 11, 13, 21: sum 72] sum

5_SUBTOTAL = 31

[4, 12, 15, 17, 21, 22, 23, 24: sum 138] sum

------- [1, 2, 5, 9, 10, 13, 14, 18: sum 72] sum

------- [1, 2, 6, 7, 11, 13, 14, 18: sum 72] sum

------- [1, 2, 6, 9, 10, 11, 14, 19: sum 72] sum

------- [1, 2, 7, 8, 9, 11, 16, 18: sum 72] sum

------- [1, 3, 5, 8, 10, 11, 16, 18: sum 72] sum

------- [1, 3, 6, 8, 10, 11, 13, 20: sum 72] sum

------- [1, 3, 7, 8, 9, 10, 14, 20: sum 72] sum

------- [2, 3, 5, 6, 10, 13, 14, 19: sum 72] sum

------- [2, 3, 5, 7, 8, 13, 16, 18: sum 72] sum

------- [2, 3, 5, 8, 9, 10, 16, 19: sum 72] sum

------- [2, 3, 5, 8, 9, 11, 14, 20: sum 72] sum

------- [2, 3, 6, 7, 8, 11, 16, 19: sum 72] sum

6_SUBTOTAL = 43

[5, 10, 15, 18, 21, 22, 23, 24: sum 138] sum

------- [1, 2, 3, 11, 12, 13, 14, 16: sum 72] sum

------- [1, 2, 4, 9, 12, 13, 14, 17: sum 72] sum

------- [1, 2, 6, 7, 11, 12, 16, 17: sum 72] sum

------- [1, 2, 6, 8, 9, 13, 16, 17: sum 72] sum

------- [1, 2, 6, 8, 11, 12, 13, 19: sum 72] sum

------- [1, 2, 7, 8, 9, 12, 14, 19: sum 72] sum

------- [1, 3, 4, 8, 11, 12, 16, 17: sum 72] sum

------- [1, 3, 6, 7, 8, 14, 16, 17: sum 72] sum

------- [1, 4, 6, 7, 9, 11, 14, 20: sum 72] sum

------- [2, 3, 4, 6, 11, 13, 16, 17: sum 72] sum

------- [2, 3, 4, 7, 9, 14, 16, 17: sum 72] sum

------- [2, 3, 4, 7, 11, 12, 14, 19: sum 72] sum

------- [2, 3, 4, 8, 9, 13, 14, 19: sum 72] sum

------- [2, 3, 6, 7, 8, 11, 16, 19: sum 72] sum

------- [2, 4, 6, 7, 8, 9, 17, 19: sum 72] sum

7_SUBTOTAL = 58

[6, 9, 14, 19, 21, 22, 23, 24: sum 138] sum

------- [1, 2, 4, 10, 11, 12, 15, 17: sum 72] sum

------- [1, 3, 4, 8, 11, 12, 16, 17: sum 72] sum

------- [1, 3, 5, 8, 10, 11, 16, 18: sum 72] sum

------- [2, 3, 5, 7, 8, 13, 16, 18: sum 72] sum

------- [2, 3, 5, 7, 10, 12, 13, 20: sum 72] sum

8_SUBTOTAL = 63

[7, 9, 12, 20, 21, 22, 23, 24: sum 138] sum

------- [1, 3, 4, 8, 10, 14, 15, 17: sum 72] sum

------- [1, 3, 4, 8, 11, 13, 14, 18: sum 72] sum

------- [1, 3, 5, 8, 10, 11, 16, 18: sum 72] sum

------- [2, 3, 4, 5, 13, 14, 15, 16: sum 72] sum

------- [2, 3, 4, 6, 11, 13, 16, 17: sum 72] sum

------- [2, 3, 4, 8, 10, 11, 15, 19: sum 72] sum

------- [2, 3, 5, 6, 10, 13, 14, 19: sum 72] sum

------- [2, 4, 5, 6, 8, 13, 15, 19: sum 72] sum

------- [3, 4, 5, 6, 8, 10, 17, 19: sum 72] sum

9_SUBTOTAL = 72

(2004.03.09: More anon.)

In 2004 I published a paper in JMP in which I motivated a new kind of spinor space

by pointing out the how exceptional were the dimensions 2, 8 and 24 (at least as they

pertain to lattice theory). This other triple of numbers (30, 36, 52) has intrigued me since

I first started playing with the idea of breaking the sequence (1,2,...,24) into

three octets, such that the sum of the squares of each octet is also a perfect square.

Interestingly:

30 = 28 + 2;

36 = 28 + 8;

52 = 28 + 24.

The chances that you have got this far in this most obscure of my pages are remote.

The chances that if you have these sums might spark some thoughts that give this

coincidence meaning are even remoter ... vanishingly remote, in fact.

It's the kind of coincidence that makes me go, "What the ...?", but from which

I find it nearly impossible to divert my gaze.

An alternative:

30 = 26 + 4;

36 = 26 + 10;

52 = 26 + 26.

Magic square, anyone?

(2007.12.04: More anon.)

Regarding the numbers 72, 90, 138 looked at above.

72 = 3*24

90 = 3*30

138 = 3*46

(24 ± 6)

Yeh, well, cool ... I think. I mean, how can that not be cool?

(2010.01.07: More anon.)