In cents, what is the least total amount that cannot be obtained by using a combination of fewer than eight coins from a collection of pennies, nickels, dimes and quarters?

94 cents

Step-by-step explanation:

Using eight coins, the maximum is dollars, so we need to find a value lower than that. If we imagine pennies, and add one to that, since we can make any value under cents, we will get cents. We can make this with a nickel, so we also need to be out of range of nickels, so we should try . We can make this with dimes, so we should try . can be made with quarters, so we should try and get cents. But, we cannot forget about subtracting from these values. If we try , we get , which is made with dimes. Trying makes , but if we keep moving it down to the nearest multiple of , or a quarter, we will get . can be made with coins though, but if we take it away again down to , we get . This looks hopeful. This has quarters, dime, nickel, and four pennies. If we keep trying and move it down to , we can still make it. If we keep moving down till we get to the LCM of the values, we get . Making this does works, so now we know it is between cents and cents. Trying the nearest multiple of minus , we get . This obviously works, so we need to get away from the multiple of . Trying , we get , , and coin types, so this works. Now try . This is quarters, dime, and four pennies. Trying , we get quarters, dime, nickel, and four pennies.There is nothing lower than this, so we have found it! :D This took me very long to do, and even I learned something from doing your super hard question! Thanks for the workout!

EDIT: Oops! I did less than or equal to! But you can use the same logic to do it again!