By Hook or By Crook

[Title from the opening of The Prisoner.]

Marika and I are back at her parents' house until we move to my next postdoc, and I've noticed something about the clock/timer on their stove. If I stand too close, the curve of the front cover blocks the tops of the digits:

Generally I'm able to read the time while only seeing the bottoms of the digits, and it got me wondering about the amount of information carried in each segment.

Digital clocks use 7-segment LEDs, which are laid out like this:

Wikipedia

We can make a table of the segments used for each digit:

	a	b	c	d	e	f	g
0	✓	✓	✓	✓	✓	✓
1		✓	✓
2	✓	✓		✓	✓		✓
3	✓	✓	✓	✓			✓
4		✓	✓			✓	✓
5	✓		✓	✓		✓	✓
6	✓		✓	✓	✓	✓	✓
7	✓	✓	✓
8	✓	✓	✓	✓	✓	✓	✓
9	✓	✓	✓	✓		✓	✓

Now we can imagine removing one or more columns from that table, and see which digits are still distinguishable. I put together a Python script to try out all the possibilities, but I struggled to find a good way to show all 126 of them. I settled on an animated GIF:

You can think of the red cells as ones that are known to be blocked, or burnt out. The list at the bottom shows the digits that remain unique. Interestingly, the lower-right cell is used in all digits except 2, so as long as that one works and is off, we know the digit is 2.

Turning to the situation above, the available digits are fairly slim:

However, since this is a clock/timer, we know the digits will be changing sequentially. For half of the possible digits, we only need to wait for at most the next one to be on a known value.