You are given n > 0 of each of the standard denomination US coins: 1¢, 5¢, 10¢, 25¢, 50¢, $1. What is the smallest n such that it is impossible to select n coins that make exactly a dollar?

For n=77, we must prove no combination of 77 coins (using at most 77 of each denomination) sums to exactly 100 cents. Working from 77 pennies (77 cents), replacing pennies with higher-value coins increases the total by 4, 9, 24, or 49 cents respectively. Every combination we test falls short of 100 cents - the closest we can get is 96 or 103 cents, but never exactly 100. For smaller values like n=67, valid combinations exist (e.g., 33 pennies, 4 nickels, 22 dimes, 8 quarters). Therefore 77 is indeed the smallest n where making exactly a dollar becomes impossible.