After many exciting adventures in Wonderland, Alice finds herself in the court of the Queen of Hearts. As she tries to sneak through the garden, she overhears the king and queen having a heated argument. “It’s quite simple,” says the queen. “64 is the same as 65, and that’s that.” Without thinking, Alice jumps in, “That’s nonsense! If 64 were the same as 65, then it would be 65 and not 64 at all.”
The queen, feeling challenged, exclaims, “How dare you! I’ll prove it right now!” Before Alice can say anything, she’s taken to a field with two chessboard patterns: an 8 by 8 square and a 5 by 13 rectangle. The queen claps her hands, and four soldiers come forward, lying down next to each other to cover the first chessboard. Alice notices that two of them are trapezoids and the other two are triangles.
“See, this is 64,” the queen says confidently. She claps her hands again, and the soldiers rearrange themselves on the second chessboard. “And that is 65.” Alice is shocked because the soldiers didn’t change size or shape. She suspects the queen is up to something sneaky.
Just when things look bad for Alice, she remembers her geometry lessons. She examines the trapezoid and triangle soldiers lying next to each other. They seem to cover exactly half of the rectangle, with their edges forming a line from corner to corner. If that’s true, the slopes of their diagonal sides should match. However, when she calculates the slopes, she finds they are different.
Before the queen’s guards can stop her, Alice drinks some of her shrinking potion to get a closer look. She discovers a small gap between the triangles and trapezoids, forming a parallelogram that accounts for the missing square.
Interestingly, the numbers involved are part of the Fibonacci series, where each number is the sum of the two preceding ones. Fibonacci numbers have special properties: first, squaring a Fibonacci number gives a value that’s one more or one less than the product of the Fibonacci numbers on either side of it. Second, the ratio between successive Fibonacci numbers is similar and converges on the golden ratio. This allows the queen to create slopes that appear deceptively similar.
In fact, the Queen of Hearts could create a similar puzzle using any four consecutive Fibonacci numbers. The higher they go, the more it seems like the impossible is true. But as Lewis Carroll, the author of Alice in Wonderland and a mathematician, noted—one can’t believe impossible things.
Recreate the queen’s trick by drawing an 8×8 square and a 5×13 rectangle on graph paper. Cut out shapes similar to the soldiers (two trapezoids and two triangles) and try to fit them into both shapes. Can you find the hidden gap? Discuss why the illusion works.
Research the Fibonacci sequence and list the first 15 numbers. Identify patterns and calculate the ratio between successive numbers. Discuss how these numbers relate to the golden ratio and how they were used in the queen’s trick.
Using a ruler and protractor, measure the angles and sides of the trapezoids and triangles from the chessboard illusion. Calculate the slopes of the diagonal sides and explain why they don’t match. What does this tell you about the shapes?
Prepare a short presentation or skit where you explain the queen’s trick to your classmates. Use props or drawings to demonstrate the illusion and reveal the hidden gap. Encourage your classmates to ask questions and solve the puzzle themselves.
Design a similar puzzle using different shapes and numbers. Try to create an illusion where the area seems to change. Share your puzzle with classmates and see if they can solve it. Discuss the mathematical principles behind your creation.
After many adventures in Wonderland, Alice finds herself in the court of the Queen of Hearts. As she tries to pass through the garden unnoticed, she overhears the king and queen arguing. “It’s quite simple,” says the queen. “64 is the same as 65, and that’s that.” Without thinking, Alice interjects, “Nonsense. If 64 were the same as 65, then it would be 65 and not 64 at all.”
“How dare you!” the queen huffs. “I’ll prove it right now!” Before Alice can protest, she is taken to a field with two chessboard patterns—an 8 by 8 square and a 5 by 13 rectangle. The queen claps her hands, and four soldiers approach, lying down next to each other to cover the first chessboard. Alice notices that two of them are trapezoids and the other two are triangles.
“See, this is 64,” the queen says. She claps her hands again, and the soldiers rearrange themselves on the second chessboard. “And that is 65.” Alice gasps, realizing the soldiers didn’t change size or shape. She suspects the queen is cheating somehow.
Just as things seem dire for Alice, she recalls her geometry and examines the trapezoid and triangle soldiers lying next to each other. They appear to cover exactly half of the rectangle, with their edges forming a line from corner to corner. If that’s true, the slopes of their diagonal sides should be the same. However, when she calculates the slopes, she finds they are different.
Before the queen’s guards can stop her, Alice drinks some of her shrinking potion to get a closer look. She discovers a small gap between the triangles and trapezoids, forming a parallelogram that accounts for the missing square.
Interestingly, the numbers involved are part of the Fibonacci series, where each number is the sum of the two preceding ones. Fibonacci numbers have properties that are relevant here: first, squaring a Fibonacci number gives a value that’s one more or one less than the product of the Fibonacci numbers on either side of it. Second, the ratio between successive Fibonacci numbers is similar and converges on the golden ratio. This allows the queen to create slopes that appear deceptively similar.
In fact, the Queen of Hearts could create a similar puzzle using any four consecutive Fibonacci numbers. The higher they go, the more it seems like the impossible is true. But as Lewis Carroll, the author of Alice in Wonderland and a mathematician, noted—one can’t believe impossible things.
Geometry – The branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, and shapes. – In geometry class, we learned how to calculate the area of different shapes.
Trapezoid – A four-sided flat shape with at least one pair of parallel sides. – The teacher asked us to find the height of the trapezoid given its area and base lengths.
Triangle – A three-sided polygon with three angles. – We used the Pythagorean theorem to find the length of the hypotenuse in a right triangle.
Rectangle – A four-sided polygon with opposite sides equal and all angles right angles. – The area of a rectangle can be found by multiplying its length by its width.
Square – A four-sided polygon with all sides equal and all angles right angles. – A square is a special type of rectangle where all sides are the same length.
Slope – The measure of the steepness or incline of a line, often represented as the ratio of rise over run. – We calculated the slope of the line by finding the change in y over the change in x.
Diagonal – A line segment connecting two non-adjacent vertices of a polygon. – The diagonal of a rectangle divides it into two congruent triangles.
Parallelogram – A four-sided polygon with opposite sides parallel and equal in length. – To find the area of a parallelogram, multiply the base by the height.
Fibonacci – A sequence of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. – The Fibonacci sequence appears in various natural patterns, such as the arrangement of leaves on a stem.
Ratio – A comparison of two quantities by division, often expressed as a fraction. – The ratio of the length to the width of the rectangle is 3:2.
