# Specific numbers and a blank box

There are 2 rows of numbers shown below. Which number belongs to the blank box with a question mark? Please choose from the three options stated.

Why were these particular numbers chosen? I think @Martund got partway there, but to move my comments on their answer from five months ago into an answer (in the name of closure)...

C - 2605

Why? Note that, as @Martund spotted, the top row of numbers all:

contain each of the five vowels of the English alphabet exactly once each when written out:

206 = twO hUndrEd And sIx;
230 = twO hUndrEd And thIrty;
250 = twO hUndrEd And fIfty;
260 = twO hUndrEd And sIxty;
602 = sIx hUndrEd And twO.
Note that this puzzle requires the use of the linking 'and' between the hundreds and tens/units, as is the convention in UK English.

However, what is crucial to the puzzle is to realise that these are:

the first five numbers to satisfy this pattern. (See OEIS A058180)

And that (related) when we look at the bottom row of numbers:

the given numbers are the first four numbers to contain each of the five vowels of the English alphabet exactly twice each when written out:

2506 = twO thOUsAnd fIvE hUndrEd And sIx;
2530 = twO thOUsAnd fIvE hUndrEd And thIrty;
2550 = twO thOUsAnd fIvE hUndrEd And fIfty;
2560 = twO thOUsAnd fIvE hUndrEd And sIxty.

i.e. This is why the OP chose these particular rows of numbers when setting the puzzle - it isn't arbitrary, and they are not seeking a forced algebraic or manipulation-of-numbers justification for the sequence.

So the next number in this sequence (denoted by the question mark) should be:

2605 = twO thOUsAnd sIx hUndrEd And fIvE. i.e. option C.

Note that neither of the other two options even appear in this sequence:
2902 = twO thOUsAnd nInE hUndrEd And twO (3 O's, only 1 I);
6052 = sIx thOUsAnd And fIfty twO (no E's, only 1 U).

It is

A because the difference between bottom and top is 2300.

It is also

B because the bottom row is obtained by inserting a 5 inside the 2.

Finally, it is

C because the bottom row is obtained by replacing 2 by 5 and afterwards prepending a 2.

Arguably, the numbers were chosen to

make the choice between A, B and C ambiguous.

• Ntnva cyrnfr pbafvqre jul jrer gurfr cnegvphyne ahzoref pubfra. V pbhyq unir pubfra nal bgure ahzoref naq nqqrq 2300 be gubfr bgure guvatf
– DrD
Mar 5, 2021 at 13:44
• @DrD not without sacrificing ambiguity. Mar 5, 2021 at 13:48

The correct option is $$B$$. Reason is that the numbers in lower table are obtained by appending $$5$$ beside $$2$$ in corresponding numbers above. (Another reason one can think is that the last entry in both the tables are obtained by reversing the first entry).

These particular numbers (206, 230, 250, 260, 602) are chosen since these are the numbers whose english names include all the vowels exactly once. Then we append $$5$$ particularly, because this the number of vowels in english.

• Please consider the question why were these particular numbers chosen.
– DrD
Mar 5, 2021 at 13:25
• @DrD, you have written that as another part of the question. My answer is partial as of now. Mar 5, 2021 at 13:26
• The problem here is that you need to demonstrate WHY your answer is the intended one, and the other ones are not the intended one - after all, it is also possible to obtain Answer A simply by adding 2300 to each number in the top row. No doubt there is also a logic that can produce Answer C (which I have not spotted yet). The statement you are dismissing as 'another part of the question' is actually crucial to solving it! :) EDIT: Your new answer now looks more promising!
– Stiv
Mar 5, 2021 at 13:29
• @Stiv, I've completed my answer. Mar 5, 2021 at 13:32
• In fact they could just be the first 5 like the top row...
– Stiv
Mar 5, 2021 at 13:47