What is the sequence?

The following 5 sequences of base-10 numbers all follow the same pattern, and converge at 14,.

What is the pattern and which number is next?

1,5,10,14,23,...
2,6,10,14,23,...
3,9,14,23,...
4,9,14,23,...
8,14,23,...

• +1 (A) Here, "Continue same way" means "1,5,10,14,1,5,10,14,1,5,10,14,...." ? (B) What about starting with "7,...." & starting with "0,...." ? (C) Should we use Decimal only ? (D) Do we use only Mathematics or will we require knowledge of something else ?
– Prem
Sep 5, 2022 at 9:00
• @Prem A) no, it means they converge at 14, - so each sequence becomes the same after 14,. I'll add the next number too, to make that clearer. All of the sequences continue increasing to infinity, ultimately. B) well, maybe I'll add that in as a hint if no-one gets there. C) what do you mean by "Decimal"? They're all integers. D) I don't want to answer that yet Sep 5, 2022 at 10:25

The rule to go from one number to the next is

add one more than the number of letters in the name of the number.

So for example, 3 is followed by 9 because

3=THREE has 5 letters, so you need to add 5+1=6 to the original 3 to get 9.

The next number:

Twenty-three has 11 letters, so the next number is 23+11+1=35.

Since each number only depends on the one directly before it, as soon as two sequences hit the same number, they will be the same thereafter.

• That was an awesome deduction. I've been hours with a paper scribbling, i'd never achieve that conclusion :P Sep 5, 2022 at 12:48
• @fontastico I started with the assumption that the each number only depends on the directly previous one, which is reasonable given the fact that the sequences start matching as soon as they share a number. The sequences only increase, ao I looked what gets added in each step, and tried to relate that to the number. From there the answer was not hard to find. Sep 5, 2022 at 13:16
• +1 , The sequence starting with 7 would have given this away sooner !
– Prem
Sep 5, 2022 at 13:56
• @Prem yeah, that's why I didn't include it. It would have come later if no-one had got it. I suppose I should have added a word-sequence tag too - that would have been another clue later on I think Sep 5, 2022 at 15:40