- The numbers on the cards may be any positive integers whatsoever.
- Carol/Rachel need not draw exactly six cards: it is only required that she draw (and, of course, put up) at least one card.
- CECIL can show any positive integer whatsoever.
- If (and only if) the number shown on CECIL matches the number on any one of the cards, no arithmetic is required.

- Carol/Rachel has complete control over which numbers are drawn.
- CECIL is operating as a counter: 1, 2, 3, 4, ..., you get the idea. CECIL keeps counting until you can no longer find a solution.

"Two numbers, please, Rachel."

"Here we go: 1 and 3"

I manage:

1 = 1

2 = 3-1

3 = 3

4 = 3+1

5 = ??? and even Rachel can't help me here.

"Rachel, were those the best numbers you could have given me?"

What is her reply?

"Three numbers, please, Rachel."

"You get 1, 2, and 6"

Well, then:

1 = 1

2 = 2

3 = 1+2

4 = 6-2

5 = 6-1

6 = 6

7 = 6+1

8 = 6+2

9 = 6+2+1

10 = (6-1)*2

11 = (6*2)-1

12 = 6*2

13 = (6*2)+1

14 = (6+1)*2

15 = ??? and it looks like I'm stuck here, too.

"Rachel, were those the best numbers you could have given me?"

What is her reply?

Similarly for four numbers, five numbers, ...

I have a feeling that this is a very difficult puzzle to solve.

If I were restricted to addition, an ideal numbers selection would be the beginning of the sequence 1, 2, 4, 8, 16, ... (doubling)

If I were restricted to addition and/or subtraction, and I am not mistaken, then an ideal numbers selection would be the beginning of the sequence 1, 3, 9, 27, 81, ... (tripling)

It seems multiplication helps: but by how much?

Is division at all useful?

If some poor soul in purgatory is assigned the task of getting CECIL up to a googol, how many cards need the numbers angel put up, and what numbers do those cards show?