Digit Cancelling Fractions

The search term for this concept is an anomalous cancellation. We can write a function that checks if an anomalous cancellation can happen by storing the digits of the numerator and denominator into four variables total, then checking four separate cases for if a digit can be "cancelled."

def digits(n):
    return (n // 10, n % 10)


def can_cancel_digits(n, d):
    x, y = digits(n)
    z, w = digits(d)

    f = QQ(n/d)

    if x == z and w != 0 and y/w == f:
        return True
    elif x == w and y/z == f:
        return True
    elif y == z and w != 0 and x/w == f:
        return True
    elif y == w and w != 0 and x/z == f:
        return True

    return False

We're only dealing with two-digit numerators and denominators, so this is easy to brute force.

fractions = set()
for n in range(10, 100):
    for d in range(n + 1, 100):
        if can_cancel_digits(n, d):
            fractions.add((n, d))

fractions

{(16, 64), (19, 95), (26, 65), (49, 98)}

There's only four such fractions.

prod(QQ(n/d) for (n, d) in fractions).denominator()

100

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