Measure my old Ruler

For 1:20-21. 2:15-17. 3:17-20. 4:17-21. 5:15-20. 6:15-21. 7:8-15. 8:0-8. 9:8-17. 10:21-31. 11:20-31. 12:8-20. 13:8-21. 14:17-31. 15:0-15. 16:15-31. 17:0-17. 18:21-39. 19:20-39. 20:0-20. 21:0-21. 22:17-39. 23:8-31. 24:15-39. 25: cannot be measured So 25 is the least length that cannot be measured

Here's simple python code that solves this problem.

l = [0,8,15,17,20,21,31,39]
p = range(40)
d = []

for i in l:
    for j in l:
        d += [abs(i-j)]

print min(list(set(p) - set(d)))

Which returns $25$ .

First set up a table of distances measurable by two marks. Start with the series visible marks. Then find the distances 1-mark away (minus the value of previous mark), 2-, 3-mark and so on.

• Visible marks: 0, 8, 15, 17, 20, 21, 31, 39
• 1-mark away: 8, 7, 2, 3, 1, 10, 8
• 2-mark away: 15 , 9, 5, 4, 11, 18
• 3-mark away: 17, 12, 6, 14, 19
• 4-mark away: 20, 13, 16, 22
• 5-mark away: 21, 23, 24
• 6-mark away: 31, 31
• 7-mark away: 39

The above are all measurable distances using the ruler. To find the shortest distance not measurable, we sort the data and we get what follows.

1 2 3 4 5 6 7 8 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 31 31 39

It can be seen that the first one missing from sequence and hence the shortest distance not measurable is $\boxed{25}$ .