# Design a ruler which can be used to measure all the lenghts from 1 unit to 12 units but this time put the minimum possible number of marks on the ruler.

mlehuzzah | Certified Educator

Super fun question!

You need at least four marks:

Suppose you had only three marks, located at A inches, A+B inches, and A+B+C inches from the left edge of the ruler (and that A+B+C+D=12)

Then the possible lengths that could be made would be:

A, B, C, D, A+B, B+C, C+D, A+B+C, B+C+D, A+B+C+D

There are only 10 numbers here, so even if they were all distinct, you couldn't make all the numbers 1-12

You need no more than four marks:

Put marks at the 1,3,5,11

-|--|--|------|-

1: 1 mark

2: space from 1 to 3

3: 3 mark

4: space from 1 to 5

5: 5 mark

6: space from 5 to 11

7: space from 5 to 12 (edge of ruler)

8: space from 3 to 11

9: space from 3 to 12

10: space from 1 to 11

11: mark at 11

12: whole ruler

Thus, the minimum number of marks needed to be able to make each of the lengths from 1 to 12 is four