Can you pay any amount?

Question

In a remote country, the monetary unit is called UNI. They only
have bills of 20 UNI and coins of 3 UNI.

You want to buy a toy at a store, but the clerk can't give
you the change back (so you have to pay the exact price).

Show that it is always possible to buy a toy as long as its price
is large enough...

Answer 1

1 UNI --> no way
2 UNI --> no way
3 UNI --> 3 UNI
4 UNI --> no way
5 UNI --> no way
6 UNI --> 2 x 3 UNI
...
19 UNI --> no way
20 UNI --> 20 UNI
21 UNI --> 7 x 3 UNI
22 UNI --> no way
23 UNI --> 20 UNI + 3 UNI
24 UNI --> 8 x 3 UNI
25 UNI --> no way
26 UNI --> 20 UNI + 2 x 3 UNI
27 UNI --> 9 x 3 UNI
28 UNI --> no way
...
34 UNI --> no way
35 UNI --> 20 UNI + 5 x 3 UNI
36 UNI --> 12 x 3 UNI
37 UNI --> no way
38 UNI --> 20 UNI + 6 x 3 UNI
39 UNI --> 13 x 3 UNI
40 UNI --> 2 x 20 UNI

Aha! Now we have 3 in a row. And if we add 3 UNI to any of them we can make 41, 42 and 43 respectively. Add 3 more and we can make 44, 45 and 46... So that is it.

As long as the price is 38 UNI or above, we can pay for it.

Answer 2

20 + (3 x 6) = 38

so u can buy a toy costing 38 UNI

3 x 13 = 39

so u can buy a toy costing 39 UNI

20 x 2 = 40

so u can buy a toy costing 40 UNI

if u can buy one for 38, u can buy one for 41 (add 3)

if u can buy one for 39, u can buy one for 42 (add 3)

if u can buy one for 40, u can buy one for 43 (add 3)

if u can buy one for 41 (38+ 3), u can buy one for 44 (add 3)

and so on..

so it needs to cost atleast 38 UNI

Answer 3

Sorry...no answer but what a great question.