hope this !
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! for the workout!
edit: oops! i did less than or equal to! but you can use the same logic to do it again!