Longest Roman Numeral
Read the puzzle
A key shows Roman letter values: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000. Examples: 27 = XXVII (5 letters) and 64 = LXIV (4 letters). Using the standard form (9 is IX, never VIIII), which number from 1 to 99 has the longest Roman numeral?
Which number from 1 to 99 has the longest Roman numeral, written the usual way?
Which digits need the most letters on their own?
88
88 = LXXXVIII, 8 letters. No other number below 100 has 8.
How to solve it
Treat the tens digit and the ones digit separately, since a numeral below 100 is just the tens part followed by the ones part.
Step 1: letter cost of each ones digit. 1 = I (1), 2 = II (2), 3 = III (3), 4 = IV (2), 5 = V (1), 6 = VI (2), 7 = VII (3), 8 = VIII (4), 9 = IX (2). The longest is 8, at 4 letters.
Step 2: the tens follow the same pattern with X, L and C. 10 = X (1), 20 = XX (2), 30 = XXX (3), 40 = XL (2), 50 = L (1), 60 = LX (2), 70 = LXX (3), 80 = LXXX (4), 90 = XC (2). The longest is 80, at 4 letters.
Step 3: combine the two longest parts. 80 + 8 = 88, written LXXX plus VIII = LXXXVIII, which is 4 + 4 = 8 letters. No other pairing reaches 8, because only the digit 8 costs 4 letters.
The usual wrong guess is 99, the biggest number, but it is XCIX, just 4 letters. Another is 38, XXXVIII, with 7.
Takeaway: length depends on the digits, not on the size of the number.
