Harrison and Lara—two tireless adventurers who face mathematical roadblocks and puzzles wherever they go—have to cross a wide river before they can continue on their current adventure.
A traveler going in the same direction tells the adventurers that the quickest and safest way over the river is a toll bridge. They agree to cross the bridge together, and split the $30 toll three ways. They each give the gatekeeper 10 one-dollar coins, totaling $30. Right away, they notice a tiny sign that says: Discount for Parties of Three or More. The three travelers demand their discount.
The gatekeeper knows that the official discount for a group is $5. He always pockets the discount when groups don't notice the tiny sign, but he also takes a cut when they do. "OK, the discount is $3," he says, dropping two of the dollar coins in his pocket and returning one dollar each to Lara, Harrison and the traveler.
So, each of the three has paid $9 for a total of $27, and the gatekeeper has kept $2. That’s $29 accounted for but where did the other dollar go? The gatekeeper didn't get it!
The flaw in the logic lies in counting the $2 kept by the gatekeeper as part of the $27 paid by the adventurers. The correct way to break down the expenses is as follows:
- Each adventurer initially paid $10, totaling $30.
- The gatekeeper took $2 and returned $1 to each adventurer, totaling $3.
- Therefore, each adventurer paid $9 ($10 - $1) and collectively they paid $27 in total.
- The remaining $2 kept by the gatekeeper is not part of the money paid by the adventurers, so it should not be added back to the $27.
In conclusion, the adventurers paid $27, the gatekeeper kept $2, and there is no missing dollar.
Let's break down the scenario step by step to understand where the other dollar went:
1. Initially, the total toll amount is $30, and Harrison, Lara, and the traveler each contribute $10, which adds up to $30.
2. The gatekeeper announces a $3 discount, taking $2 for himself and returning $1 to each of them.
3. At this point, Harrison, Lara, and the traveler have each paid $9, which totals to $27 ($9 x 3 = $27).
4. The $2 taken by the gatekeeper is not included in the $27 paid by the adventurers since the adventurers are only concerned with their individual payments.
5. Therefore, the sum of the adventurers' payments ($27) and the gatekeeper's earnings ($2) gives us the total amount of $29 ($27 + $2 = $29).
There is no missing dollar. The apparent discrepancy in the calculation arises because adding the $2 taken by the gatekeeper to the amount paid by the adventurers does not make sense since they are separate transactions.