You have 1000 coins. Start by placing one coin in its own "stack." Next to the first stack, make a stack of two coins. Then make a stack of four coins and continue creating stacks of coins, each stack twice as high as the previous one.

Now I figured it would be 13 stacks by doing it out longhand.
What formula would you use to find this?

that is the sum of a geometric series

1000 = sum from n = 1 to n = n of
a + ar + a r^2 + a r^3
where a = 1
and r = 2
the sum of the first n terms is
a (1-r^n) /(1-r)
so here
1000 = 1 (1-2^n)/(1-2)
-1000 = 1 - 2^n
2^n = 1001
n log 2 = log 1001
n = 3/.301 = about 10

I don't think 10 is correct.

I did it long hand
1,2,4,8,16,32,64,128,256,512,1024,2048,
4096 = 13 stacks.

Your terms are

1, 2 , 4 , 8
or
1 ,2^1 , 2^2 , 2^4 ...... 2^12

that sum is
2^n = 10,001
or
n (.301) = 4
n = 4/.301 = 13

In other words you either have a typo and you mean 10,000 coins, not 1,000 coins, or I am very confused

notice the following pattern

2^0 = 1
2^0 + 2^1 = 3 = 2^2 - 1
2^0 + 2^1 + 2^2 = 7 = 2^3 - 1 , 3 terms in sum
2^0 + 2^1 + 2^2 + 2^3 = 15 = 2^4 - 1 , 4 terms in sum
...
2^0 + 2^1 + ... + 2^9 = 1023 = 2^10 - 1

(the sum of n powers of 2 starting with 2^0 or 1, will be 2^(n+1) - 1

so 11 terms give you a sum of 1023
and 10 terms give you a sum of 511

So as Damon calculated,
9 stacks are not enough, using only 511 coins,
and 10 stacks will leave the last stack short since you need 1023 to make 10 stacks

To find the number of stacks, you can use a formula that relates the total number of coins to the height of the highest stack in terms of powers of 2.

Let's break down the problem. You start with one coin in the first stack, then two coins in the second stack, four coins in the third stack, and so on. Each stack has twice as many coins as the previous stack. So the pattern is: 1, 2, 4, 8, 16, and so on.

To determine the height of the highest stack, you need to find the largest power of 2 that is less than or equal to the total number of coins. In this case, you have 1000 coins.

The formula you can use to find the number of stacks is:

n = log2(N)

Where:
- n is the number of stacks.
- N is the total number of coins.

In this case, N is 1000. To calculate n, you can take the logarithm of 1000 to the base 2.

n = log2(1000)

Using a calculator or a logarithmic table, you can find that log2(1000) is approximately 9.97.

Since the number of stacks must be a whole number, you would round up the value of n to the nearest integer. In this case, you would round up to 10.

Therefore, using the formula, you would find that it takes approximately 10 stacks to reach or exceed 1000 coins.