(i) How many patterns does he need to form the smallest square? (ii) What are…

(i) How many patterns does he need to form the smallest square? (ii) What are the dimensions of the largest square that he can form?

I can’t give the numeric answers yet — I need the full problem statement (how many patterns he has and the size/shape of each pattern).

Explanation / what I need

  • Please tell me whether the “patterns” are identical tiles, and give their dimensions (or area), and whether the supply is limited (how many patterns he actually has).
  • Without that, there are several common interpretations; below I show the standard methods so you can apply them once you supply the missing data.

If the question is: “How many identical rectangular patterns (each of size a×b) are needed to make the smallest square?”

  • Let the area of one pattern be $m=a\\times b$.
  • You need the smallest integer $N$ such that $N\\cdot m$ is a perfect square.
  • Factor $m=\\prod p_i^{e_i}$. For each prime $p_i$ with odd exponent $e_i$, include one extra $p_i$ in $N$ to make the exponent even.
  • Equivalently, write $m=u^2\\cdot t$ where $t$ is squarefree; then the minimal $N$ is $t$.
  • Example: if a pattern is $2\\times3$ so $m=6=2\\cdot3$, then $t=6$ and the smallest $N$ is $6$ (since $6\\cdot6=36$, a perfect square). The resulting square will have area $36$ and side length $6$ (if units are unit squares).

If the question is: “Given a finite number $T$ of identical patterns, what is the largest square (by number of patterns) he can form?”

  • The largest number of whole patterns you can arrange into a perfect square count is the largest perfect square ≤ $T$, i.e. $\lfloor\\sqrt{T}\\rfloor^2$ patterns.
  • The side length (in patterns) is $\lfloor\\sqrt{T}\\rfloor$.
  • Example: if he has $T=20$ patterns, the largest square uses $\\lfloor\\sqrt{20}\\rfloor^2=4^2=16$ patterns, arranged as a $4\\times4$ square of patterns.

If you paste the exact problem (pattern dimensions and how many he has), I’ll compute (i) the minimal number needed for the smallest square and (ii) the dimensions of the largest square he can form, with full step-by-step working.

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