Enigma 1698: The pied piper
From New Scientist #2865, 19th May 2012 [link]
I found an old eight-digit calculator. As usual, each position of the display consisted of seven light segments – with, for example, seven lighting up to display an “8” and four lighting up to display a “4”.
To see if it still worked, I typed in the approximation for 𝜋, namely 22/7, and found that I saw a number, but not the correct one. After some investigation I found that one of the 56 segments always failed to light. I then used 22/7 again to calculate approximations for 2𝜋, 3𝜋, 4𝜋, 5𝜋, 6𝜋, 7𝜋, 8𝜋, 9𝜋 and 10𝜋.
In two-thirds or more of these cases the answer looked like a number, but:
(a) how many of them were correct? and
(b) what was the original eight-digit number seen in the display?