# Just a little sum-thing

Select 5 numbers. Using exactly five copies of each, simultaneously create 15 (multi-)sets, one of each sum from 1 to 15 inclusive.

Thanks to Gordon Hamilton for the inspiration...

• Despite a successful solution (though perhaps not the only one possible!), it seems that some confusion remains. Could you please let me know what needs clarification, so that I can improve the puzzle? Thanks... – Zomulgustar Dec 24 '18 at 6:28

We must select 5 numbers with sum $$\frac{1}{5}(1+2+\dots+15) = \frac{1}{5} \cdot 120 = 24.$$

Select $$1, 3, 5, 7, 8.$$ Here is the list:

$$1$$

$$1, 1$$

$$3$$

$$1, 3$$

$$5$$

$$3, 3$$

$$7$$

$$8$$

$$8, 1$$

$$5, 5$$

$$8, 3$$

$$7, 5$$

$$8, 5$$

$$7, 7$$

$$8, 7$$

Motivation: $$1$$ is forced. I tried adding $$2$$ next, but that would've forced me to use a number above $$8$$ ($$\min \max = 9$$ achieved with $$\{1, 2, 4, 8, 9\}$$), so I added $$3$$ to the list. The only 5 element set with $$1, 3$$ and $$\max \le 8$$ was $$\{1, 3, 5, 7, 8\},$$ and it worked.

• Nicely done! The solution is unique if you restrict to natural numbers. – Zomulgustar Dec 23 '18 at 21:09