cold
cord
word
worm
warm

5+4=9 (remove stick from the 6 and add to the 4 to create a 9)
8-4=4 (remove stick from the plus sign and create an 8 from the 6)

a possible solution? 1, 2, 4, 8 and 16 grams. Please correct if you find this to be wrong.

Why?

We want to create the numbers from 1-31 by adding up only 5 numbers.

we do need 1.
we might need 2.
we can create 3 from 2 and 1. (1+2=3)
we cannot create 4 from 1 and 2 so we need to use 4.
we can create 5 from 1 and 4 (1+4)
etc.

if we repeat this we can arrive to the number 15 using  only 1, 2, 4, 8

Now we can stop since we know that 16+15=31. That means we can create all remaining numbers from 17 to 31 by adding 16+ all the numbers from 1-15

1
2
3 (1+2)
4
5 (4+1)
6 (4+2)
7 (4+2+1)
8
9 (8+1)
10 (8+2)
11 (8+2+1)
12 (8+4)
13 (8+4+1)
14 (8+4+2)
15 (8+4+2+1)
16
17 (16+1)
18 (16+2)
31 (16+15) or (8+4+2+1)