What is the smallest number of coins (excluding 50 cent piece) thats value can sum to any amount .01 to .99?
This is a question that I came up with today and my immediate thought is 3Q, 2D, 1N, 4P.
My thought are, how do you justify such a thing? Are there more than one solution?
Let's begin narrowing it down with this: no amount can ever be further than 4 from the nearest X5 or X0. What does this mean? The answer is automatically at most, for .99, 10 dimes and a nickel and 4 pennies which would be 15. And you can subtract one from this answer for each digit below 9 for the tens place in .99...but let's consider the impact of quarters. For .99, this would be 3 quarters, 2 dimes, and 4 pennies...an answer of 9. What we see now, is that any other significant scenario (i.e. 94, 89, 69, 64, etc.) is in some way analogous to the case of .99 and results either in the same number of coins or fewer (i.e. with .94 just replace one of the dimes in .99 with a nickel and you get the same result of nine coins; and with .84 or .89 just remove a dime from .94 and .99 respectively and both answers yield results of eight coins). >Thus the answer is nine.