MathDB

I have two cents and Bill has

n

cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a dime for a dozen pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend all

n

of his cents on some combination of pencil packages to get

P

pencils. However, if I give my two cents to Bill, he then notes that he can instead spend all

n+2

of his cents on some combination of pencil packages to get fewer than

P

pencils. What is the smallest value of

n

for which this is possible?
Note: Both times Bill must spend all of his cents on pencil packages, i.e. have zero cents after either purchase.

Problem Statement