It all started with a 1 smart kid
A few weeks ago, Patrick Honner, who is an award-winning math teacher, posted a realization on Twitter. His 7-year-old had just realized discovered that their 300-piece jigsaw puzzle was made out of 324 pieces as it was in an 18 x 18 format.
Calculating number of pieces – DIY
What this means is that a puzzle number is regarded as valid if it is a y x z format. In this format, y is lesser than or equal to z and z is equal to or lesser than 4z. The constant 4 is a random choice and can be any number according to the specification of a certain puzzle. With that in mind, a puzzle number would be something that looks like this, 10x4 = 40, 20x4 = 80, and 30x4 = 120. In the OEIS puzzle, numbers can be found under A071562, where they are described as numbers whose middle divisor is not zero. A middle divisor is the divisor of a number that is between the square root of a number divided by 2 and the square root of a number x 2. This means your middle divisors will have to be 8.660254037844386 and 34.64101615137755 to make a puzzle that will have exactly 300 pieces. However, puzzle piece rows are rarely ever arranged in this manner as they usually get rounded over to the nearest natural number. That is why most manufacturers choose to divide their desired number of pieces with a middle divisor to determine how many pieces will be on one row.
truth vs. Practice
After determining how many rows one side will have using the middle divisor, the remaining number will be used to constitute the other side’s rows. Due to this, the number of pieces advertised on jigsaw puzzle boxes rarely ever represent what you will find inside and math enthusiasts on the internet have realized this.
One of those math enthusiasts is known as John D. Cook once said jigsaw puzzles that claim to have 1,000 pieces mean they approximately have 1,000 pieces. He said the term “1000-piece” is not meant in its literal form because puzzle pieces are normally arranged in a grid-like formation. Since the pieces are in a grid-like formation, this means the number of pieces on one side is a divisor of the total number of pieces. Cook said the grid formation found in many jigsaw puzzles makes it very hard for manufacturers to make a puzzle that has exactly 1,000 pieces. Cook’s assessment of 1,000-piece jigsaw puzzles makes it easier for us to understand why most puzzles have aspect ratios that produce numbers that are around the advertised sum.
Also, there is a puzzle blog that suggests that most 500-piece jigsaw puzzles have 513 pieces. The blog says this because most manufacturers use an aspect ratio of 27 x 19 to make 500-piece puzzles. They also said that most manufacturers also use 38 x 27 aspect ratios to make 1,000-piece puzzles. The 38 x 27 and 27 x19 aspect ratios can be translated as “2Y-piece” and “y-piece.” That makes it a better working model for manufacturers who want to produce both 500-piece and 1,000-piece puzzles.