Jigsaw Puzzle Riddle

Number Theory Level 4

John Spilsbury likes jigsaw puzzles and is a mathematics riddler.

He states: "I only own landscape oriented puzzles in which the number of border pieces is exactly $8\%$ of the total number of pieces in that puzzle."

From each possible ( 'landscape' oriented) jigsaw puzzle that can be manufactured according to the given restriction John has one such puzzle in his small collection.

What is the total number of pieces of all of John's puzzles combined?

Note : Corner pieces also belong to the border.

The answer is 22200.

1 solution

Maria Kozlowska
May 8, 2015

Let $a, b$ be number of rows and number of columns, so $a > b$ . Number of border pieces is $2 a + 2b - 4$ so we have $\frac{2 a + 2b - 4}{ab} = 0.08$ $b=25+25\frac{23}{a-25}$ $\Rightarrow a$ can have only following values: $50, 140, 600$ as number $a-25$ has to divide $25 \times 23$ and $a > b$ . The total number of pieces is then: $50\times 48 + 140 \times 30 + 600 \times 26 = \boxed{22200}$

Great job Maria! Thank you for this solution; hope you enjoyed the problem.

For some rewriting $\frac{2a + 2b - 4}{ab} = 0.08$ to your final expression of $b$ may not be so straightforward and you might have left out some algebraic steps yourself to make the solution more compact. I like that as it may still be a challenge for anyone who 'looks' at your solution.

Additional relevant question to other Brilliant-users :

How can we quickly determine the number of valid solutions to this problem (3 in this case) by analyzing the number 575 (Maria wrote $25 \times 23$ )?

Patrick Heebels - 6 years, 1 month ago

Thanks for your kind comment Patrick.

Maria Kozlowska - 6 years, 1 month ago

Jigsaw Puzzle Riddle

The answer is 22200.

1 solution

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