I'm writing a series of number labels and I need to ensure that the label is not read upside down so I underline any label that could be read upside down and if doing so would give a different number than reading it right way up. As I'm doing so I wonder, how many labels will need the underline?
Note:
A number can be read upside down if and only if it consists of only 0, 1, 6, 8 and 9. Leading zeroes are allowed in the upside down versions (so 1000 can be read upside down).
So how many labels do I have to underline when my series goes from 001 to 999? What about to 9999? Is there a general answer for a series ending at 10^N - 1?
EDIT: Although I did mention it, it should be properly stated that numbers will be padded with leading zeros to all have N digits (the same N as in the 10^N - 1) and that labels which when read upside down are the same do not need underlines (so if going to 999, 111 doesn't need an underline but 011 does).
EDIT2: The real thrust of the question will follow once the preliminary answers have been worked out.
Originally posted by pawnfondlerYou're forgetting that numbers that read the same either way up don't need an underline. The rest of your logic seems fine.
so any number in the 200,300,400,500 and 700 sections would be out? as well as any number with 2 3 4 5 and 7 in it. so would i be correct by saying find out how many numbers can be upside down in a single hundred section and multiply by 5? and same would apply for the 000-9999?
Originally posted by XanthosNZprobably.
Is there a general answer for a series ending at 10^N - 1?
The number of candidates for underlining is 5^N - 1
Then we need to weed out those that are symmetrical, like 010, 818, 619 which should be formulaic
for N=3 symmetricals = (5x3)-1
for N=4 symmetricals = (5x5)-1
for N=5 symmetricals = (5x5x3)-1
for N=6 symmetricals = (5x5x5)-1
and I guess
for N=7 symmetricals = (5x5x5x3)-1
for N=8 symmetricals = (5x5x5x5)-1
for N=3 underlines = (5^3)-1 - ((5x3)-1) = 124 - 14 = 110
for N=4 underlines = (5^4)-1 - ((5x5)-1) = 624 - 24 = 600