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While strolling home through the industrial district late one night with a puzzle stuck in your head, you wander off course into an area you’ve never been before. As you turn in the direction you think you need to go, two bright machines catch your eye. At first you think they are gambling machines, but upon closer inspection you find a sign which explains it all.

Did you know there are other values of coins besides the ones you are used to? (This is the U.S. so there are coins/bills worth \$0.01, \$0.05, \$0.10, \$0.25, \$0.50, \$1.00, \$5.00, \$10.00, \$20.00) We here at Infinimint deal with coins of all possible values. We prefer to work with the common values, so if it is possible to give you a value which is normal (like a dime, nickel, etc) the machine will do just that.

The two vending machines in front of you have their rules listed on the front:

Insert one coin. Subtract one cent then divide it into three equal-valued coins (Rounded down to the nearest cent).

Insert two coins. Add their values together minus one cent.

You think oh that’s interesting, but it's getting late, I should head home. On your way out, a shiny object on a table catches your eye. You pick it up. It says across the top \$24.99. “Well this is useless”, you think to yourself. No one is going to accept this ridiculous coin for payment. But, you think, maybe there is some way I can get real money out of the vending machines. I should be able to get \$20 easy, maybe even more.

What is the most money in normal values, \$0.01, \$0.05, \$0.10, \$0.25, \$0.50, \$1.00, \$5.00, \$10.00, \$20.00, that can be made from the \$24.99 coin?

I'm not sure what the answer is, but I think it will be apparent if someone has proven that their solution is best.

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    $\begingroup$ Two new tags just for one puzzle? $\endgroup$
    – greenturtle3141
    Commented Feb 25, 2017 at 2:09
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    $\begingroup$ @greenturtle3141 I tried entering tags, but no suggestions showed up. I would change though if there's a better idea for what to use. $\endgroup$
    – Tony Ruth
    Commented Feb 25, 2017 at 6:05
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    $\begingroup$ Mathematics? Calculation? $\endgroup$
    – greenturtle3141
    Commented Feb 25, 2017 at 6:09
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    $\begingroup$ The accepted answer is incorrect. The math and calculations are wrong at early stages, conditioning the whole answer. $\endgroup$
    – Alvaro Montoro
    Commented Mar 1, 2017 at 23:59

2 Answers 2

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EDIT: Improved answer thanks to Alvaro Montoro

Now I took liberty to assume that it's perfectly valid to use some of your own coins because if the objective is to make to most normal money out of the invalid money that is the way to go I think and with this I can go from $24.99 to

Only 4 cents loss

And I do it like this:

\$24.99 + \$5.00 - \$0.01 = \$29.98
\$29.98 + \$0.05 - \$0.01 = \$30.02
(\$30.02 - \$0.01) / 3 = 3 * \$10.00 rounded down
each step only causes 1 cent loss and 1 from rounding down so it's only 4 cents loss

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  • $\begingroup$ I like this answer. If you had a coin worth $5.03 you could do it even more efficiently. $\endgroup$
    – Tony Ruth
    Commented Feb 27, 2017 at 15:24
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I believe I can get $24.60

Divide 24.99 in 3, get 3 coins of 8.32
Add two of the coins back together, get a coin worth 16.61
Divide an 8.92 coin in 3, get 3 coins of 2.77
Add one of these coins to the 16.61 coin, total value of 19.37
Divide one of these coins in three, getting three coins of 92 cents each
Divide two of these coins in three, yielding six coins of 30 cents each Add one of these coins to the large coin, giving a total of 19.66
Divide two of the five remaining 30 cent coins in three, yielding six coins worth nine cents each
Add four of the six to the largest coin, giving a total of 19.98
Split one of the remaining 9 cent coins into three 2 cent coins
Add two of the two cent coins to the largest coin, giving a 20.00 coin. Add the last 2 cent coin to the 9 cent coin, yielding a 10 cent coin.

First step complete:

At this point, we have a 20 dollar coin, a 2.77 coin, a 92 cent coins, three 30 cent coins, and a ten cent coin. We can combine the 2.77 coin with the 92 cent coin to get a 3.68 coin To this we can add the three 30 cent coins to get a 4.55 coin We can split this for three $1.51 coins, which we can split for nine 50 cent coins.

Total: $24.60

We now have one coin worth 20, nine coins worth .50 each, and one coin worth .10

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    $\begingroup$ The first addition is incorrect, making the calculation off by 2 cents: 8.32 + 8.32 - 0.01 = 16.63 $\endgroup$
    – Alvaro Montoro
    Commented Feb 25, 2017 at 0:36

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