蛮力解决方案返回了 Project Euler 问题 31 的错误答案

Question

I'm having trouble with Project Euler problem 31 . The correct answer is 73,682, and my answer is 73,681. Here is the problem:

In the United Kingdom the currency is made up of pound (£) and pence (p). There are eight coins in general circulation: 1p, 2p, 5p, 10p, 20p, 50p, £1 (100p), and £2 (200p).

It is possible to make £2 in the following way: 1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p.

How many different ways can £2 be made using any number of coins?

And my code:

counter = 0
for a in range(3):
    for b in range(5):
        if 100 * a + 50 * b > 200:
            break
        for c in range(11):
            if 100 * a + 50 * b + 20 * c > 200:
                break
            for d in range(21):
                if 100 * a + 50 * b + 20 * c + 10 * d > 200:
                    break
                for e in range(41):
                    if 100 * a + 50 * b + 20 * c + 10 * d  + 5 * e > 200:
                        break
                    for f in range(101):
                        if 100 * a + 50 * b + 20 * c + 10 * d + 5 * e + 2 * f <= 200:
                            counter += 1
print(counter)

Answer 1

The problem is that your code doesn't consider the case where you just have a £2 coin. You can just initialize the variable counter to 1 to fix this issue.