What Do You Know About Hinged Dissections?

1) At which point are the pieces of a hinged dissection hinged?

Center

Left

Right

Vertices

The pieces of a hinged dissection are hinged at the vertices. This means that the points where the individual pieces connect and rotate are located at the vertices of the shapes involved in the dissection.

Explanation

The pieces of a hinged dissection are hinged at the vertices. This means that the points where the individual pieces connect and rotate are located at the vertices of the shapes involved in the dissection.

2) What makes a twist hinged dissection different from a regular hinged dissection?

It is plane

It is three-dimensional

Its hinges are placed on the vertices

It allows overlapping

A twist hinged dissection is different from a regular hinged dissection because it is three-dimensional. While a regular hinged dissection occurs in a plane, a twist hinged dissection exists in three dimensions. This means that the objects being dissected can be manipulated and reconfigured in three-dimensional space, allowing for more complex and intricate transformations.

Explanation

A twist hinged dissection is different from a regular hinged dissection because it is three-dimensional. While a regular hinged dissection occurs in a plane, a twist hinged dissection exists in three dimensions. This means that the objects being dissected can be manipulated and reconfigured in three-dimensional space, allowing for more complex and intricate transformations.

3) What connects each points together in a hinged dissection?

Dots

Hinges

Locks

Chains

A hinged dissection is a geometric puzzle where a shape can be divided into smaller pieces that are connected by hinges. These hinges allow the pieces to move and be rearranged into different configurations. Therefore, the correct answer is "hinges" as they are the connecting element in a hinged dissection.

Explanation

A hinged dissection is a geometric puzzle where a shape can be divided into smaller pieces that are connected by hinges. These hinges allow the pieces to move and be rearranged into different configurations. Therefore, the correct answer is "hinges" as they are the connecting element in a hinged dissection.

4) Who introduced the idea of hinged dissections?

Greg Frederickson

Jones Lewis Mulberry

Henry Ernest Dudeney

George Rodriguez

Henry Ernest Dudeney is the correct answer for introducing the idea of hinged dissections. Dudeney was a famous English mathematician and author who specialized in mathematical puzzles and recreations. He is known for his contributions to recreational mathematics, particularly his work on dissection puzzles. Dudeney's interest in hinged dissections led him to explore and create various puzzles and solutions involving the rearrangement of polygons by hinging them along certain edges. His work in this area has had a significant impact on the field of recreational mathematics.

Explanation

Henry Ernest Dudeney is the correct answer for introducing the idea of hinged dissections. Dudeney was a famous English mathematician and author who specialized in mathematical puzzles and recreations. He is known for his contributions to recreational mathematics, particularly his work on dissection puzzles. Dudeney's interest in hinged dissections led him to explore and create various puzzles and solutions involving the rearrangement of polygons by hinging them along certain edges. His work in this area has had a significant impact on the field of recreational mathematics.

5) At what point are the hinges placed on the pieces in a twist hinge?

Vertices

Edge

Center

Bottom

In a twist hinge, the hinges are placed on the edges of the pieces. This allows the pieces to rotate or twist around the edge, providing flexibility and movement. Placing the hinges on the edges ensures that the pieces can pivot smoothly and maintain stability while twisting. Therefore, the correct answer is "Edge."

Explanation

In a twist hinge, the hinges are placed on the edges of the pieces. This allows the pieces to rotate or twist around the edge, providing flexibility and movement. Placing the hinges on the edges ensures that the pieces can pivot smoothly and maintain stability while twisting. Therefore, the correct answer is "Edge."

6) Which of these shapes is formed by joining unit squares at their edges?

Polygons

Polyforms

Polyominoes

Rhombi

Polyominoes are formed by joining unit squares at their edges. A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. The term "polyomino" is derived from the Greek words "poly" meaning "many" and "omino" meaning "square". Each individual square in a polyomino is called a unit square. Therefore, polyominoes are the correct answer to the question.

Explanation

Polyominoes are formed by joining unit squares at their edges. A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. The term "polyomino" is derived from the Greek words "poly" meaning "many" and "omino" meaning "square". Each individual square in a polyomino is called a unit square. Therefore, polyominoes are the correct answer to the question.

7) Which of these is not a way to form a new shape with a hinged dissection?

Overlays

Adding a new piece

Moving the piece

Folding

Adding a new piece is not a way to form a new shape with a hinged dissection. Hinged dissection involves rearranging and repositioning the existing pieces of a shape to form a new shape, without adding any new pieces. This process relies on the flexibility of the hinged connections between the pieces. Adding a new piece would not be considered a hinged dissection, as it involves introducing a completely new element to the shape transformation.

Explanation

Adding a new piece is not a way to form a new shape with a hinged dissection. Hinged dissection involves rearranging and repositioning the existing pieces of a shape to form a new shape, without adding any new pieces. This process relies on the flexibility of the hinged connections between the pieces. Adding a new piece would not be considered a hinged dissection, as it involves introducing a completely new element to the shape transformation.

8) Which of these is true about two polygons that have a common hinged dissection?

Both must have the same area

One should be twice the size of the other

One must be half the size of the other

Number of sides of both polygons must be less than ten

If two polygons have a common hinged dissection, it means that they can be dissected into the same set of smaller polygons that can be rearranged to form either of the original polygons. In order for this to be possible, both polygons must have the same area. If one polygon was twice the size of the other, it would not be possible to rearrange the smaller polygons to form both of them. Similarly, if one polygon was half the size of the other, the smaller polygons would not be able to form both of them. Therefore, the correct answer is that both polygons must have the same area.

Explanation

If two polygons have a common hinged dissection, it means that they can be dissected into the same set of smaller polygons that can be rearranged to form either of the original polygons. In order for this to be possible, both polygons must have the same area. If one polygon was twice the size of the other, it would not be possible to rearrange the smaller polygons to form both of them. Similarly, if one polygon was half the size of the other, the smaller polygons would not be able to form both of them. Therefore, the correct answer is that both polygons must have the same area.

9) Which of these does not occur when you form a new shape in a dissection?

The original shape will be changed

The chain will be broken

The original shape will remain intact

The chain will not be broken

When forming a new shape in a dissection, the chain will be broken. This means that the original arrangement or sequence of connected components will be disrupted or disconnected. This is because dissection involves rearranging or separating the parts of a shape to create a new configuration or arrangement. The other options state that the original shape will be changed, the original shape will remain intact, and the chain will not be broken, but these are incorrect as they contradict the nature of dissection.

Explanation

When forming a new shape in a dissection, the chain will be broken. This means that the original arrangement or sequence of connected components will be disrupted or disconnected. This is because dissection involves rearranging or separating the parts of a shape to create a new configuration or arrangement. The other options state that the original shape will be changed, the original shape will remain intact, and the chain will not be broken, but these are incorrect as they contradict the nature of dissection.

10) What was the first hinged dissection?

Equilateral triangle to square

Equilateral triangle to rectangle

Equilateral triangle to polygon

Equilateral triangle to hexagon

The first hinged dissection refers to the first known transformation of an equilateral triangle into a square using only hinged movements. This means that the triangle can be rearranged by folding along certain points without any cuts or additions. The equilateral triangle to square transformation is considered the first hinged dissection because it was the earliest discovered and documented example of this type of geometric transformation.

Explanation

The first hinged dissection refers to the first known transformation of an equilateral triangle into a square using only hinged movements. This means that the triangle can be rearranged by folding along certain points without any cuts or additions. The equilateral triangle to square transformation is considered the first hinged dissection because it was the earliest discovered and documented example of this type of geometric transformation.