Did you solve it? A tray of Portuguese delights

Berlin Balls or Bola de Berlim for sale in a cake shop window in Porto
Photograph: Alamy Stock Photo

Earlier today I set you four puzzles by Paulo Ferro, a Portuguese puzzle maker.

1. Trapezium or trap-not-so-easy-um?

The trapezium is composed of 7 matchsticks. Move the positions of 3 matchsticks in order to obtain 2 equilateral triangles.

(The final arrangement must use all the matchsticks, and have no extraneous lines.)

Solution:

2. Nine times table

The number 403 is created out of 15 matchsticks. Move the positions of two matchsticks to obtain a number divisible by 9.

For example, two from the first digit moved to the third digit give you 108. Find the other four solutions.

Solution

3. Touching squares

Two squares meet at the centre of a circle. The light blue square has area 20.

What is the area of the dark blue square?

Solution 12.5

Here’s how to solve it with Pythagoras’s theorem. The red lines marked r are the radius of the circle. The side of the light blue square is √20 = 2√5.

From the right-angled triangle in light blue square that has hypotenuse r, we get:

(√5)² + (2√5)² = 5 + 20 = 25 = r².

So r = √25.

If a is the side length of the dark black square, we get a² + a² = 2a² = r² = 25. Thus a² = 12.5, and a² is also the area of the square.

4. Pent-up thoughts

A regular pentagon and a square have the same side length.

What is the size of the angle?

(You will need to know that the internal angles of a triangle, a quadrilateral and a pentagon add up to 180, 360 and 540 degrees respectively.)

Solution: 162 degrees

The internal angles of a pentagon are 108 degrees, and of a square a 90 degrees. Thus the angle marked in the triangle above must by 108-90 = 18 degrees. The two lines that meet at this angle are of equal length (because the square and pentagon have same side length) and thus the triangle that they make is isosceles. Since the angles of a triangle add up to 180 degrees, the other two angles marked must be (180-18)/2 = 81 degrees.

We deduce that the left angle of the light blue triangle above must be 108-81 = 27 degrees. Since we know the top angle is 108 degrees, the angle on the right must be 180 - 108 – 27 = 45 degrees.

The dark blue quadrilateral is symmetric, and thus the left angle must be 45 degrees. The remaining angle – the answer – is thus 360 – 45 -45 – 108 = 162 degrees.

If you enjoyed these brainteasers, you may also enjoy Paulo’s new book book of geometry puzzles, Birds, Bees and Burgers.

Thanks to Paulo Ferro for today’s puzzles. You can follow him on Twitter. His website is EnigMaths.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book.