Posted by ColinWright 10/26/2025
Yes the script lets you check that the result is correct, but a proof lets you see why it's correct. A good proof might even give you a sense of how you could have discovered the result yourself, or how you might generalize it.
pp = lambda x : denom(x)/ (num(x) - denom(x)*(x - 2))
[pp(2),pp(4),pp(6),pp(8)]
[1.0, 9.0, 373.0, 48913.0]
11 * 11 = 121
111 * 111 = 12321
1111 * 1111 = 1234321
and so on, where the largest digit in the answer is the number of digits in the multiplicands.
That test wouldn't detect a dead left side on the 2nd from-right digit
Calculator displays are multiplexed, so the usual defects are either one digit that never displays anything, or one segment that stays blank on all digits.
The defect mentioned by you is frequent only on displays with independent digits (like some digital clocks), not on calculators.
I do not know whether on calculator LCD displays there are frequent cases when a single segment can become defect.
At the time about which I am talking, calculators had either green vacuum fluorescent displays (like mine) or red LED displays. With such displays, the normal defects were either in the driving circuits or in the connections to the multiplexed display, so they affected either all segments of a digit or the same segment in all digits. I have never seen a case when the actual light-emitting segment of a digit of a VFD or LED display was defect.
Pick an integer between 1 and 9. Multiple it by 9. Take that number and multiply it by 12345679. (Skip the 8)
>>> 3 * 9
27
>>> 12345679 * 27
333333333
This all works because:
>>> 111111111 / 9
12345679.0
Base 3: 21/12 = 7/5(dec.)
Base 2: 1/1 = 1
Base 1: |/| = 1 (thinking |||| = 4 etc.)