The Bases of the Probabilities

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If you have 26 letters, and if you add these to each other, how many possibilities do you have for numeric outcomes?

First combination of two is, A + A = 2 (in "gematria by 1") or 2+2*64 = 130 in ASCII. Next combination of two is value 3 or rather 131, you have two possibilities, either AB or BA. Midway you will have 26 possibilities for AZ, MN, NM, ZA and so on. At ZZ you are down to one sole possibility for 180, again.

1		130		180
2		131		179
3		132		178
4		133		177
5		134		176
6		135		175
7		136		174
8		137		173
9		138		172
10		139		171
11		140		170
12		141		169
13		142		168
14		143		167
15		144		166
16		145		165
17		146		164
18		147		163
19		148		162
20		149		161
21		150		160
22		151		159
23		152		158
24		153		157
25		154		156
26		155

First combination of three is, A+A+A = 3 or rather 3+3*64 = 195. It's one only. Next, 196, has three possibilities, AAB, ABA, BAA. One, then three, seems like triangle numbers? Well, yes, to a point.

You cannot get higher in triangle numbers than 1+2+3 ... +26, that is 351. Because next value will also only have 26 possibilities, like one for each letter of the third, and to the former two the diverse combinations, ranging from 2 to 26, which makes 25 of the letters, and then 25 only for Z. Next after that will have 3 to 26, then 25 for Y and 24 for Z. And so on.

Now, if triangle number of 26 is 351, next value will be 351 - 1 + 25, which is 375. Next value after that will be 375 - 2 + 24, which is 397. You may notice that if first after triangle numbers differs by - 1 + 25, it differs concretely by + 24. Next differs by - 2 + 24, concretely by + 22. This means, the values get higher and higher, but by decreasing increments, up to 501 + 4 = 505, 505 + 2 = 507. And after that, next one is also 507, then 505, 501 and back to 351 and the triangle numbers down to 1.

Here are he triangle numbers part, increasing from 1 for 195 and then decreasing to 1 for 270:

1		195		270
3		196		269
6		197		268
10		198		267
15		199		266
21		200		265
28		201		264
36		202		263
45		203		262
55		204		261
66		205		260
78		206		259
91		207		258
105		208		257
120		209		256
136		210		255
153		211		254
171		212		253
190		213		252
210		214		251
231		215		250
253		216		249
276		217		248
300		218		247
325		219		246
351		220		245

So far triangle numbers, and now for decreassing increments:

375		221		244
397		222		243
417		223		242
435		224		241
451		225		240
465		226		239
477		227		238
487		228		237
495		229		236
501		230		235
505		231		234
507		232		233

Any combination of three upper case letters of the normal English alphabet will be between 195 and 270, and its probability will be between 1 possibility for each of these two to 507 possibilities for each of 232 and 233.

Note, it is impossible to know how many of the letter combinations give meaning in any language and how many of those adding up to a particular number give meaning in any one. But the proportions in all combinations including gibberish will indicate something about the proportions in meaningful combinations.

Hans Georg Lundahl
Cergy
St. Nicolas of Tolentino
10.IX.2019