Rod measures
Problem
Using 3 rods of integer lengths, none longer than 10 units and not using any rod more than once, you can measure all the lengths in whole units from 1 to 10 units. How many ways can you do this?
For example with rods of lengths $3, 4, $ and $9$ the measurements are:
$4-3,$ $9-4-3,$ $3,$ $4,$ $9-3,$ $9-4,$ $3+4,$ $9+3-4,$ $9,$ $9+4-3,$
Using 3 rods of ANY integer lengths, what is the greatest length N for which you can measure all lengths from 1 to N units inclusive? Can you beat 10 units? Can you beat the highest value of N submitted to date?
Getting Started
There are exactly 9 different solutions for N=10 units with no rod longer than 10 units. Can you find them? With one of these solutions you can measure up to N=13. Can you do better if you can choose 3 longer measuring rods?
If you double the lengths of all three rods in one of these solution sets you can measure even lengths from 2 to 2N units. Introducing a one unit rod to make up a set of four measuring rods you can now measure all lengths from 1 to (2N+1) units. Can you do better than 27 units with 4 rods?
Student Solutions
There was a great deal in this problem and James has answered part of it:
Using only three rods with each one not exceeding 10 you can add and minus the following numbers which can go from 1-13.
The three numbers are 9, 3 and 1.
| 1 | = | 1 |
| 2 | = | 3-1 |
| 3 | = | 3 |
| 4 | = | 3+1 |
| 5 | = | 9-3-1 |
| 6 | = | 9-3 |
| 7 | = | 9-3+1 |
| 8 | = | 9-1 |
| 9 | = | 9 |
| 10 | = | 9+1 |
| 11 | = | 9+3-1 |
| 12 | = | 9+3 |
| 13 | = | 9+3+1! |
Well done James. Is 13 the largest number and why?
Can any of you help with some of the rest of the question:
In how many ways can you make 10?
What is the maximum number you can make with 4 rods?
Perhaps you can send in some more ideas.
Teachers' Resources
Rod Measures
Using 3 rods of integer lengths, none longer than 10 units and not using any rod more than once, you can measure all the lengths in whole units from 1 to 10 units. How many ways can you do this?
For example with rods of lengths $3, 4, $ and $9$ the measurements are:
$4-3,$ $9-4-3,$ $3,$ $4,$ $9-3,$ $9-4,$ $3+4,$ $9+3-4,$ $9,$ $9+4-3,$
Using 3 rods of ANY integer lengths, what is the greatest length N for which you can measure all lengths from 1 to N units inclusive? Can you beat 10 units? Can you beat the highest value of N submitted to date?