Number The Doors

In this puzzle, submitted by Jean Marie Fade, we invite you to find some unusual and interesting ways to number 20 things in sequence.

The Schlumberger center in Angola has 20 offices that each need to have a number placed on the door.

The office manager will purchase metal numerals and nail them to the doors. The cost of the numerals varies depending on the digit:

Numeral		Cost

1		1 Kwanza
2		2 Kwanzas
3		3 Kwanzas
4		4 Kwanzas
5		5 Kwanzas
6		6 Kwanzas
7		7 Kwanzas
8		8 Kwanzas
9		9 Kwanzas
0		10 Kwanzas

So, for example, putting the number 6 on a door would cost 6 Kwanzas. The number 13 would cost 4 Kwanzas. The number 10 would cost 11 Kwanzas.

The only requirement is that the numbers be in consecutive order. What is the least expensive way to do the job?

Course:

Math [1]

Result/Solution(s)

If the doors are numbered in the conventional way—1 through 20—the cost is 122 Kwanzas (Kw).

Solution - one possibility

We can save 2 Kw by starting the numbering at 0 instead of 1 because we get rid of the 20, which costs 12 Kw, and add the 0, which costs 10 Kw.

Solution - one possibility

Bernard Montaron suggests that we try a different base. Base 2, using only 1s and 0s would be very expensive because of all the zeros at 10 Kw each, but using base 7 costs only 104 Kw.

Solution - one possibility

It also turns out that base 8 numbering costs the same as using base 7, but base 6 or base 9 costs more.

combination problems [2]