Geometry - Puzzle Prime

Worm in an Apple

Section: Brain Teasers Category: Mathematics

Difficulty: Hard Tags: Geometry

There is a perfectly spherical apple with a radius 50mm. A worm has entered the apple, made a tunnel of length 99mm through it and left. Prove that we can slice the apple in two pieces through the center, so that one of them is untouched by the worm.

SOLUTION

Let the entering point is A, the leaving point is B and the center of the apple is C. Consider the plane P containing the points A, B and C and project the worm’s tunnel on it. Since 99 < 2×50, the convex hull of the tunnel’s projection will not contain the center C. Therefore we can find a line L through C, such that the tunnel’s projection is entirely in one of the semi-planes of P with respect to L. Now cut the apple with a slice orthogonal to P passing through the line L and you are done.

Unknown Author February 1, 2024

0 Comments

A String Around a Rod

Section: Brain Teasers Category: Mathematics

Difficulty: Medium Tags: Geometry

A string is wound around a circular rod with circumference 10 cm and length 30 cm. If the string goes around the rod exactly 4 times, what is its length?

SOLUTION

Imagine the circular rod is actually a paper roll and the string is embedded inside the paper. When we unroll it, we get a paper rectangle 30cm×40cm with the string embedded along the diagonal. Using the Pythagorean theorem, we find that the length of the string is 50cm.

Unknown Author October 7, 2023

0 Comments

FEATURED

Lost In the Forest

Section: Brain Teasers Category: Mathematics

Difficulty: Expert Tags: Featured, Geometry

You are lost in the middle of a forest, and you know there is a straight road exactly 1 km away from you, but not in which direction. Can you find a path of distance less than 640 m which will guarantee you to find the road?

SOLUTION

Imagine there is a circle with a radius of 100 m around you, and you are at its center O. Let the tangent to the circle directly ahead of you be t. Then, follow the path:

Turn left 30 degrees and keep walking until you reach the tangent t at point A for a total of 100×2√3/3 meters, which is less than 115.5 meters.
Turn left 120 degrees and keep walking along the tangent to the circle until you reach the circle at point B for a total of 100×√3/3 which is less than 58 meters.
Keep walking around the circle along an arc of 210 degrees until you reach point C for a total of 100×7π/6 which is less than 366.5 meters.
Keep walking straight for 100 meters until you reach point D on the tangent t.

Unknown Author September 22, 2023

3 Comments

FEATURED

Eye of the Dragon

Section: Brain Teasers Category: Mathematics

Difficulty: Advanced Tags: Featured, Geometry

Divide the circle below in two pieces. Then, put the pieces together to get a circle with a dragon, such that the dragon’s eye is at the center of the new circle.

SOLUTION

The solution is shown below.

Unknown Author July 25, 2023

0 Comments

Cut the Pizza

Section: Brain Teasers Category: Mathematics

Difficulty: Hard Tags: Geometry

Cut a circular pizza into 12 congruent slices, such that exactly half of them contain crust.

Remark: We say that a slice contains crust if it shares an arc with the boundary of the pizza (with non-zero measure).

SOLUTION

First, cut the pizza into 6 congruent circular triangles, and then split each of them in half, as shown on the image below.

Unknown Author July 13, 2023

4 Comments

FEATURED

Beautiful Geometry 1

Section: School Category: Mini Course

Tags: Featured, Geometry

Since 2018, Catriona Shearer, a UK teacher, has been posting on her Twitter various colorful geometry puzzles. In this mini-course, we cover some of her best problems and provide elegant solutions to them. Use the pagination below to navigate the puzzles, which are ordered by difficulty.

Example Problems

Which area is larger, the yellow or the red one?

What fraction of the outer circle is shaded?

If the circle radius is 1, what area is shaded?

Puzzle Prime April 25, 2023

0 Comments

Rapunzel and the Prince

Section: Brain Teasers Category: Mathematics

Difficulty: Medium Tags: Geometry

The evil witch has left Rapunzel and the prince in the center of a completely dark, large, square prison room. The room is guarded by four silent monsters in each of its corners. Rapunzel and the prince need to reach the only escape door located in the center of one of the walls, without getting near the foul beasts. How can they do this, considering they can not see anything and do not know in which direction to go?

SOLUTION

The prince must stay in the center of the room and hold Rapunzel’s hair, gradually releasing it. Then, Rapunzel must walk in circles around the prince, until she gets to the walls and finds the escape door.

Puzzle Prime August 1, 2021

2 Comments

Borromean Rings

Section: Brain Teasers Category: Mathematics

Difficulty: Hard Tags: Geometry

Borromean rings are rings in the 3-dimensional space, linked in such a way that if you cut any of the three rings, all of them will be unlinked (see the image below). Show that rigid circular Borromean rings cannot exist.

SOLUTION

Assume the opposite. Imagine the rings have zero thickness and reposition them in such a way, that two of them, say ring 1 and ring 2, touch each other in two points. These two rings lie either on a sphere or a plane which ring 3 must intersect in four points. However, this is impossible.

Unknown Author April 9, 2020

0 Comments

Square Cake

Section: Brain Teasers Category: Mathematics

Difficulty: Medium Tags: Geometry

There is a square cake at a birthday party attended by a dozen people. How can the cake be cut into twelve pieces, so that every person gets the same amount of cake, and also the same amount of frosting?

Remark: The decoration of the cake is put aside, nobody eats it.

SOLUTION

Divide the boundary of the cake into twelve equal parts, then simply make cuts passing through the separation points and the center. This way all tops and bottoms of the formed pieces will have equal areas, and also all their sides will have equal areas. Since all pieces have the same height, their volumes will be equal as well.

Unknown Author January 10, 2020

0 Comments

Four Points in the Plane

Section: Brain Teasers Category: Mathematics

Difficulty: Advanced Tags: Geometry

Find all configurations of four points in the plane, such that the pairwise distances between the points take at most two different values.

SOLUTION

All 6 configurations are shown below: a square, a rhombus with 60°-120°-60°-120°, an equilateral triangle with its center, an isosceles triangle with 75°-75°-30° and its center, a quadrilateral with 75°-150°-75°-150°, and a trapezoid with base angles of 72°.

Unknown Author December 7, 2019

2 Comments