Answer
- Each time you join two blocks (no matter their sizes), you reduce the total number of separate blocks by exactly 1.
- You start with \(n\) separate pieces (so \(n\) separate “blocks”).
- To end with exactly 1 block, you must perform exactly \(n-1\) joins.
- The puzzle’s rules ensure you always join two existing blocks at each step (rather than, say, adding one piece at a time), but that does not change the fact that each step reduces the number of blocks by exactly 1.
Hence the total number of “fitting” steps required is \(n - 1\).
Work Step by Step
Below is a typical strong‐induction argument that shows exactly \(n-1\) “fitting steps” are needed to assemble \(n\) jigsaw puzzle pieces into one complete block.
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## Statement
> **Claim.** For a jigsaw puzzle with \(n\) pieces, if you start by joining two matching pieces into a single block and then, at each subsequent step, join two previously assembled blocks (each consisting of one or more pieces), the total number of steps required to end up with a single block of all \(n\) pieces is \(n - 1\).
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## Proof by Strong Induction
**Base Case(s).**
1. **\(n=1\)**: There is only one piece, so there is nothing to “fit together.” The number of steps required is \(0\), and \(0 = 1 - 1\).
2. **\(n=2\)**: You simply match the two pieces. That is exactly 1 step, which equals \(2 - 1\).
Thus the statement holds for \(n=1\) and \(n=2\).
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**Inductive Hypothesis.**
Assume for some integer \(n \ge 2\) that **for every integer \(k\) with \(1 \le k \le n\)**, any jigsaw puzzle of \(k\) pieces can be assembled into a single block in \(k - 1\) steps (under the described rules).
We must prove it for \((n+1)\) pieces.
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**Inductive Step: Case of \((n+1)\) Pieces.**
1. **Partition the \((n+1)\) pieces into two groups.**
In the process of assembling, at some step you join two blocks, one of size \(i\) (containing \(i\) pieces) and one of size \((n+1-i)\) (containing the other pieces).
- By the strong inductive hypothesis, the \(i\)‐piece block can be formed in \(i-1\) steps.
- Similarly, the \((n+1-i)\)‐piece block can be formed in \((n+1-i) - 1 = n - i\) steps.
2. **Join the two blocks.**
After these two blocks are formed (in a total of \((i - 1) + (n - i) = n - 1\) steps), you perform **one more step** to fit these two blocks together into a single block of size \((n+1)\).
3. **Total step count.**
The total number of steps is
\[
\underbrace{(i-1)}_{\text{to assemble block of size }i}
\;+\;
\underbrace{(n - i)}_{\text{to assemble block of size }(n+1 - i)}
\;+\;
\underbrace{1}_{\text{to join these two blocks}}
\;=\;
n.
\]
But we need exactly \(n\) steps to form \((n+1)\) pieces into one block, i.e., \((n+1)-1\). Notice \(n = (n+1) - 1\). So indeed, it takes \((n+1) - 1\) steps.
Thus, by strong induction, any jigsaw puzzle of \(n\) pieces can be assembled in exactly \(n - 1\) steps.
