What coins and how many of each are required to pay for any item from 1 cent to $1 included without receiving any change using the least number of...

You must have at least 4 pennies: this will cover items
from 1-4 cents.

After that, you need a nickel: you now have
items from 1-9 cents covered.

After this: you need a dime:
you now have items from 1-19 cents covered.

Let's think
about items between 20-24 cents. Adding a quarter would be too much, so we'll add
another dime. We now have 1-29 cents covered.

Since we've
surpassed 25 cents, we can now add a quarter. We now have everything between 1-54 cents
covered.

To cover the rest, we could add a half
dollar.

Putting all the above together, we find the minimum
required coins to have exact price between 1 cent and 1 dollar is 4
pennies, 1 nickel, 2 dimes, 1 quarter, and 1 half
dollar.