Did you solve it? We are (puzzle) family

Photograph: Michael Ochs Archives/Redferns

Earlier today I set you the following two puzzles:

1. The two sisters

Twin girls are born in March, they celebrate their birthdays in September, when they grow up they marry each other.

Explain, please.

Solution: The girls are from March, Cambridgeshire. They become registrars who officiate at each other’s nuptials.

2. The four sisters

A mother with four daughters sews a flag in the shape of an irregular quadrilateral, which she divides by the two diagonals into four different coloured triangles, as shown below.

The areas of the triangles (in square inches) correspond exactly to the ages of the four daughters. The elder sisters are two, six and twelve years older than the youngest.

What are the ages of the daughters?

(An irregular quadrilateral just means that the sides and angles are not all the same, as they are in, say, a square.)

Solution 3, 5, 9 and 15 years old

Let the triangles have areas A, B, C and D. And let the length of the line from the bottom left corner to the centre be X, and the length from the centre to the top right corner be Y.

The red and blue triangles have the same height (marked as h). And thus the ratio of their bases (X and Y) is equal to the ratio of their areas:

X/Y = D/C.

The same reasoning with the green and yellow triangles gives X/Y = A/B.

Thus A/B = D/C, or A x C = B x D.

We can also deduce that D = A + 2, B = A + 6, and C = A + 12.

Thus A x (A+ 12) = (A + 6) x (A + 2)

A² + 12A = A² + 8 A + 12

4A = 12

A = 3, so D = 5, B = 9, C = 15.

I hoped you enjoyed today’s puzzles. I’ll be back in two weeks.

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.

Thanks to Chris Sealey for the ‘two sisters’ riddle, and to Manfred Pietsch for the ‘four sisters’ puzzle.