uncomputable / expressive-wallet

What is a minimal wallet of coins that can express all amounts below a target?

Expressive Wallet Problem

Trigger warning: math ⚠️

"Given a set of coin denominations, what is a minimal wallet that can express all amounts up to a target amount?"

This math problem actually has some use in daily life! Always pay the bill exactly ✅

Example: Euro cents 1, 2, 5, 10, 20, 50

Here are minimal wallets for targets 0 to 100.

It is surprising how slowly the wallet grows with respect to the target. I can't help but see a skyscraper.

Target	Minimal Wallet
0	[]
1	[1]
2	[1, 2]
3	[1, 2]
4	[1, 2, 2]
5	[1, 2, 2]
6	[1, 2, 2, 5]
7	[1, 2, 2, 5]
8	[1, 2, 2, 5]
9	[1, 2, 2, 5]
10	[1, 2, 2, 5]
11	[1, 2, 2, 5, 10]
12	[1, 2, 2, 5, 10]
13	[1, 2, 2, 5, 10]
14	[1, 2, 2, 5, 10]
15	[1, 2, 2, 5, 10]
16	[1, 2, 2, 5, 10]
17	[1, 2, 2, 5, 10]
18	[1, 2, 2, 5, 10]
19	[1, 2, 2, 5, 10]
20	[1, 2, 2, 5, 10]
21	[1, 2, 2, 5, 10, 20]
22	[1, 2, 2, 5, 10, 20]
23	[1, 2, 2, 5, 10, 20]
24	[1, 2, 2, 5, 10, 20]
25	[1, 2, 2, 5, 10, 20]
26	[1, 2, 2, 5, 10, 20]
27	[1, 2, 2, 5, 10, 20]
28	[1, 2, 2, 5, 10, 20]
29	[1, 2, 2, 5, 10, 20]
30	[1, 2, 2, 5, 10, 20]
31	[1, 2, 2, 5, 10, 20]
32	[1, 2, 2, 5, 10, 20]
33	[1, 2, 2, 5, 10, 20]
34	[1, 2, 2, 5, 10, 20]
35	[1, 2, 2, 5, 10, 20]
36	[1, 2, 2, 5, 10, 20]
37	[1, 2, 2, 5, 10, 20]
38	[1, 2, 2, 5, 10, 20]
39	[1, 2, 2, 5, 10, 20]
40	[1, 2, 2, 5, 10, 20]
41	[1, 2, 2, 5, 10, 20, 20]
...	...
60	[1, 2, 2, 5, 10, 20, 20]
61	[1, 2, 2, 5, 10, 20, 20, 50]
...	...
100	[1, 2, 2, 5, 10, 20, 20, 50]