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How do the number of edge pieces and the number of interior pieces compare in cases where either m ≤ 4 or n ≤ 4 ? Show that if the number of edge pieces is equal to the number of interior pieces, then m = 4 + 8 / (n - 4) - Leaving Cert Mathematics - Question Question 1
Question Question 1
How do the number of edge pieces and the number of interior pieces compare in cases where either m ≤ 4 or n ≤ 4 ? Show that if the number of edge pieces is equal to... show full transcript
Worked Solution & Example Answer:How do the number of edge pieces and the number of interior pieces compare in cases where either m ≤ 4 or n ≤ 4 ? Show that if the number of edge pieces is equal to the number of interior pieces, then m = 4 + 8 / (n - 4) - Leaving Cert Mathematics - Question Question 1
Step 1
How do the number of edge pieces and the number of interior pieces compare in cases where either m ≤ 4 or n ≤ 4?
Answer
To compare the number of edge and interior pieces in an m x n jigsaw, we must understand how each is calculated:
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Edge Pieces Calculation: The edge pieces are calculated from the perimeter of the jigsaw. Given an m x n jigsaw:
- Total edge pieces = 2m + 2(n - 2) = 2m + 2n - 4.
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Interior Pieces Calculation: The interior pieces are the total pieces minus the edge pieces. Thus:
- Total pieces = mn, and thus
- Interior pieces = mn - (2m + 2n - 4) = mn - 2m - 2n + 4.
When considering cases where either m ≤ 4 or n ≤ 4, we note that the edge pieces typically remain constant or low, making the interior pieces more numerous, especially when m and n are both small. Therefore:
- For m = 1 or 2, edge pieces count is very low.
- For n = 1 or 2, similarly, edge pieces will also drop, turning the comparison in favor of more interior pieces.
Step 2
Show that if the number of edge pieces is equal to the number of interior pieces, then m = 4 + 8 / (n - 4).
Answer
Starting with the relationship between edge and interior pieces:
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Set the Edge and Interior Pieces Equal:
2m + 2n - 4 = mn - 2m - 2n + 4
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Rearranging:
mn - 4m - 4n + 8 = 0
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Rearranging further gives us a quadratic in m:
m(n - 4) = 8 + 4n
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Solving for m gives:
m = 4 + rac{8}{n-4}.
Step 3
Find all cases in which number of edge pieces is equal to the number of interior pieces.
Answer
From the condition derived in part (b):
m = 4 + rac{8}{n - 4}.
When determining cases, we can explore integer values for n:
- If n = 5: m = 4 + 4 = 8
- If n = 6: m = 4 + 2.67 ~ 6.67 (Not allowed since m must be an integer)
- Continuing this analysis for n = 7, 8...
Possible pairs (m, n) where m and n yield integers can be explored based on this derived equation. We discover specific integer solutions, such as (8, 5) or (12, 8).
Step 4
Determine the circumstances in which there are fewer interior pieces than edge pieces. Describe fully all such cases.
Answer
To assess when interior pieces are fewer than edge pieces, we analyze:
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Condition:
mn - (2m + 2n - 4) < 2m + 2n - 4
Rearranging gives us:
mn < 4m + 4n - 4. -
Factoring the situation allows us to consider small dimensions:
- If both m and n are less than or equal to 4, calculate cases with examples of m=1, n=2 etc.
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We ultimately find limits on dimensions, suggesting there are numerous cases when both dimensions remain small, leading to relatively high edge pieces compared to limited interior pieces, specifically for m, n both <= 4.