MathDB

Let

S

be a subset of

\{0,1,2,\dots ,9\}

. Suppose there is a positive integer

N

such that for any integer

n>N

, one can find positive integers

a,b

so that

n=a+b

and all the digits in the decimal representations of

a,b

(expressed without leading zeros) are in

S

. Find the smallest possible value of

|S|

.
Proposed by Sutanay Bhattacharya
[hide=Original Wording] As pointed out by Wizard_32, the original wording is:
Let

X=\{0,1,2,\dots,9\}.

Let

S \subset X

be such that any positive integer

n

can be written as

p+q

where the non-negative integers

p, q

have all their digits in

S.

Find the smallest possible number of elements in

S.

3