Largest range of consecutive integers that can be formed

Question

Using a subset of a choice with 10 pieces from ${1, 2, 5, 10, 20, 50, 100}$, what's the largest range of consecutive integer values that can be produced (note that $0$ isn't counted)?
For example, I could take ten ones to get a range of $1$ to $10$, which would give me an answer of $10 - 1 = 9$. I can't seem to do any better than $9$. Can someone please help me with this problem? I don't think that ten ones is the answer because it seems too simple.

Jeremy Dover · Answer

I'm pretty sure this is the maximum, but without proof:

Albert Chen · Answer

1~9: at most 3 pieces. 8, 9 need 3.
10~90: at most 3 pieces. 80, 90 need 3.
so you can get from 1 to 499 without a problem.
because at most 4 x 100, 3 for 10~90, and 3 for 1~9.
But above 500, you need 5 x 100. you have 5 left for the rest.
with 5, you can get from 1 to 87.
so the range of the consecutive number is 1 ~ 587. You can't get 588 because 80 needs 3, 8 needs 3.
Edit: As people suggested in the comments, this method doesn't use a fixed set of pieces. Instead, it gets a separate set for each number.

Largest range of consecutive integers that can be formed

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