c4countdown

I was wondering about number picks for a "generalized" version of the Countdown numbers game. What I mean by "generalized" is:

• 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.

For this puzzle, we have:

• 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.

Here we go:
"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?

Robert Lozyniak wrote:Is division at all useful?

Yes. For example, with the numbers 2 2 6, you can get all the numbers 1-8, but you can't get 1, 3, 5 or 7 without division.

Great puzzle Robert!

I reckon it wouldn't be too hard to program it for 5 or 6 numbers but not long before you run into trouble from the sheer volume of numbers.

I reckon, for now, I can push the record for 3 numbers up to 15 using 2, 3 and 7.

I do this in my head ALL THE TIME, except I use four or five numbers and see how far into the three figures I can get. I am exceptional husband material.

You've stumbled upon my cure for insomnia. Unfortunately it works very well so I can't give you any amazing insights, but you can use 3 numbers to get up to 17 if you pick 2,3,10. I suspect this is the maximum possible.

Robert Lozyniak wrote: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?

One, and it shows a googol.