Place the numbers

Consider the set of positive integers.

Each integer can be definitively placed either above the line or below it.

Given the initial collection of placements below, what are

• The next two numbers in the series that should appear above the line, and
• The next two numbers in the series that should appear below the line?
• Why?

1 3 5 7 8 9 10 11 ...
---------------------
2 4 6 30 32 ...


The numbers above the line all contain an "e" in their English representation. (one, three, five, seven, eight, nine, ten, eleven, ...)

The numbers below the line don't. (two, four, six, thirty, thirty-two, ...)

Therefore, 12 and 13 should be the next numbers above the line, and 34 and 36 should be the next two numbers below.

• This looks right! Only catch is, '12, 13' and '34, 36' are the 'obvious' answers. The OP should have asked for the next three above and below. – Rand al'Thor Nov 18 '14 at 18:50
• Well, the only difference there would be that 38 was skipped over. The number after 13 is still 14, all the way up to 29. – Joe Z. Nov 18 '14 at 18:51
• I like this one; it is a good example of lateral thinking to approach the logic. – KeithS Nov 18 '14 at 19:29
• Exactly what I had in mind. Though I suppose I could have said "Consider ONLY the set of positive integers"... Nice work, Joe. – JRAnsley Nov 18 '14 at 23:49

The next two numbers on top of the line would be

12 and 13. All positive integers (natural numbers) can be placed on the line, the ones already in place are in order, and the next one after 6 below the line is 30, therefore 12 and 13 must be above the line otherwise they'd already have been placed below it.

As far as below the line, the only system I see is

taking sets of three consecutive even numbers. So the next number would be 34.

The trick is, what's the number after that? Well,

2, 4, and 6 are the first, second and third even numbers. Then 30, 32 and 34 are the 15th, 16th and 17th. So the pattern is, take three, skip eleven. The next number below the line, by this system, would be 34+24 = 58.