Fridqay Puzzler -- More clamps

Re: Fridqay Puzzler -- More clamps

Larry Barrett

I heard the CarTalk show where C&C talked about their answer and the letter they received, so I will leave that part of your Puzzler to others.

But here is my solution to the extra credit question.

Suppose you want to end up with n clamps after you leave the guard at gate 7. This means that when you arrived at gate 7 you had to have 2(n-1) clamps. You kept (n-1), gave the guard (n-1) and the guard gave you 1 back, so you left with n.

Now go back to gate 6. You would have had to arrive at gate 6 with 2[2(n-1)-1] clamps. Keep 2(n-1)-1, give the guard 2(n-1)-1, get 1 back, and you leave with 2(n-1).

The pattern now looks like this:

Arrive at gate 7 with 2(n-1) = 2n-2 = 2^1n - 2^1.

Arrive at gate 6 with 2[2(n-1)-1] = 4n-4-2 = 2^2n - 2^2 - 2^1.

If you trust your extrapolation instincts, this means you

Arrive at gate 5 with 8n-8-4-2 = 2^3n - 2^3 - 2^2 - 2^1,

...

Arrive at gate 1 with 2^7n - 2^7 - 2^6 ... - 2^1.

If you remember HS (maybe college) math, the last terms can be rewritten as

(2^8 - 2), so the general form of the number of clamps that you need when you arrive at gate 1, in order to leave gate 7 with n, is

2^7n - (2^8 - 2).

For example, if you want to leave gate 7 with 3 clamps, you need to arrive at gate 1 with 128*3 - (256-2) = 130.

Correct for extra credit!

Alex Y

You found the key to solution -- to work backwards. The "guard function" at each gate is g(n) = n/2 +1. But that is only defined for even numbers, and it is hard to find the domain of repeating that function 7 times. But the inverse of the guard function (call it "f") is easier to work with. f(x) = (x-1)*2 is defined for any positive integer.

As you pointed out, repeating this function 7 times tells you what you need to start with to come out with any particular x at the end

I originally read your answer on my phone, and didn't recognize it, since we had two very different looking versions of exactly the same answer. I remembered my answer of 128*(n-2) + 2, and didn't realize ours were the same answers until I got back to some paper and pencils!