Introduction / Context:
This puzzle is a fun combination of algebra and story telling. You are carrying cakes and must cross seven bridges, and at each bridge a troll takes half your cakes but then gives one cake back. Despite these repeated losses and gifts, you want to end up with exactly two cakes when you reach your grandmother. The question asks how many cakes you must start with. Puzzles like this illustrate how working backwards can simplify what looks like a complicated sequence of operations.
Given Data / Assumptions:
- There are exactly seven bridges to cross.
- At each bridge, the troll first takes half of your current cakes.
- After taking half, the troll always returns one cake to you.
- The process is identical at every bridge.
- You want to arrive at your grandmother house with exactly two cakes.
Concept / Approach:
The key concept is reverse working. Instead of following the story forward from the start, start at the end with the known final number of cakes and move backward bridge by bridge. At each step in reverse, you invert the operations of the troll. In the forward direction the troll takes half and then gives one, so in the reverse direction you first remove the gift of one and then double the remaining cakes. By repeating this reverse operation seven times, you can find the initial number of cakes you must carry from home.
Step-by-Step Solution:
Step 1: Let the number of cakes you have just after crossing the last bridge be 2, since you want to arrive with two cakes.
Step 2: Reverse the troll action. In reverse, before the troll gave you 1 cake, you must have had 2 minus 1 equal to 1 cake.
Step 3: Before the troll took half of your cakes, you must have had double that amount. So before the last bridge event, you had 1 * 2 = 2 cakes.
Step 4: Notice that this reverse calculation took you from after bridge 7 back to before bridge 7, and the number is still 2.
Step 5: Repeat the same reverse logic for bridge 6. After bridge 6 you would have 2 cakes. Before the gift, that is 1 cake, and before the halving, that is again 2 cakes.
Step 6: Continue this pattern backward across all seven bridges. Each time you reverse the operations, you end up with 2 cakes before the bridge.
Step 7: After stepping back through all seven bridges in this way, you find that at the very start of the journey you still need only 2 cakes.
Verification / Alternative check:
To verify, run the story forward using 2 cakes as the starting number. Before bridge 1 you have 2 cakes. The troll takes half (1), leaving 1, and then gives 1 back, returning you to 2. You cross bridge 2 with 2 cakes and the same thing happens. This repeats at each of the seven bridges. After every bridge you still have 2 cakes. At the end of the seventh bridge, you arrive at your grandmother house with exactly two cakes, which confirms that starting with 2 works perfectly.
Why Other Options Are Wrong:
If you start with 4, then at the first bridge half is taken (2), one is returned (3), and now you carry 3 cakes into the next bridge. Repeating the process will not conveniently bring you back to exactly 2 at the end. Similar mismatches occur for starting values of 6 or 8. Only 2 remains stable under the operation of halving and then adding 1 through multiple repetitions, which is why the backward reasoning leads you directly to that value.
Common Pitfalls:
A common mistake is to try to simulate the process forward with an unknown starting number, which can become confusing. Algebraic equations can be used, but they often seem heavy for a story puzzle. Working backwards is cleaner and more intuitive in this context. Another pitfall is to misinterpret half of the cakes if the number becomes odd. In this puzzle, starting with 2 avoids such issues because you always remain at 2 after each operation, making the scenario consistent and easy to track.
Final Answer:
You must start the journey with
2 cakes in order to arrive with exactly two cakes after crossing all seven bridges.
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