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; | 9-4; | 9-3, 3+4; | 9+3-4; | 9 and 9+4-3 | (as illustrated). |
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?
What is the greatest length that can be measured using 4 rods in this way? Can you beat the best solution submitted so far? Is your answer best possible?
There was a great deal in this problem and James Fletcher 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 next month.