I drew four right-angled triangles. The hypotenuse of my first triangle was also the shortest side of my second triangle; the hypotenuse of my second triangle was also the shortest side of my third triangle; the hypotenuse of my third triangle was also the shortest side of my fourth triangle. The length in millimetres of each side of each triangle was an integer less than 100.

What were the lengths of the shortest and the longest sides that I drew?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 15 January. The Editor鈥檚 decision is final. Please send entries to Enigma 1779, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1773 Cutting corners: The other two sides of the triangle were 7.0 and 7.5 centimetres

The winner Tony Griffin of Etobicoke, Ontario, Canada

Whether you write dates in the order day.month.year or month.day.year (in each case with two digits for each element), 04.04.01 represents 4 April 2001, and it is a square date because 40401 is the square of 201.

In addition to 4 April 2001, there is one other date in this century that is written in exactly the same way in each order and is also a square date. 1) What is that date? 2) What is the next date after today that is a square date if written in the order day.month.year? 3) What is the next date after today that is a square date if written in the order month.day.year? Give each answer in the form of 4 April 2001.

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 13 November. The Editor鈥檚 decision is final. Please send entries to Enigma 1771, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1765 Repeating digits: The sum of the numbers is 689

The winner Dan Sattary-Javid of Reigate, Surrey, UK

I have listed in random order five positive integers, four of two digits and one of one digit. They use each of the digits 1 to 9. None of the two-digit integers has any factor greater than 1 in common with any other of the two-digit integers. The first of the integers in my list is a triangular number. The sum of the first two, the sum of the first three, the sum of the first four and the sum of all five are also triangular numbers. What are my integers in the order in which I have listed them?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 9 October. The Editor鈥檚 decision is final. Please send entries to Enigma 1766, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1760 Squares and cubes: The numbers are 324 and 16

The winner Bryan Butler of Greytown, New Zealand

On a sheet of paper divided by horizontal and vertical lines into 1cm x 1cm cells I drew a circle whose centre was at a point where lines intersected and whose radius was an integral number of centimetres (less than 50). I counted the number of cells that the circumference of my circle passed through.

The circumference of a circle whose radius was 1cm smaller would have passed through a greater number of cells. How many cells did the circumference of my circle pass through?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 21 August. The Editor鈥檚 decision is final. Please send entries to Enigma 1759, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1753 Divisible in reverse: The numbers are 32946 and 52486

The winner Josh Kopp of Chagrin Falls, Ohio, US

I have drawn two triangles. The lengths of their sides are six different single-digit integers. The triangles have one angle in common. What are the lengths of the sides of the two triangles?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 31 July. The Editor鈥檚 decision is final. Please send entries to Enigma 1756, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1750 Navigating the grid: There are 32 numbers in the list

The winner Paul Young of Yarrawarrah, New South Wales, Australia

You have two problems to solve. Neither of your answers may have a leading zero.

1) What is the 5-digit positive integer divisible by 17 and 19 whose reverse is a larger integer also divisible by 17 and 19?

2) What is the 5-digit positive integer divisible by 7 and 23 whose reverse is a larger integer also divisible by 7 and 23?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 10 July. The Editor鈥檚 decision is final. Please send entries to Enigma 1753, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1747 Mind your Ps and Qs: P=3 and Q=4

The winner Graham Holmes of Barham, Suffolk, UK

Harry and Tom each chose two four-digit perfect squares (with no leading zero) which used eight different digits. Each of them told me which two digits were not used in his squares, and that information enabled me to deduce with certainty which two squares each of them had chosen. They had not made the same choice of squares, but their choices did have one square in common.

What was the square that both of them chose?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 22 May. The Editor鈥檚 decision is final. Please send entries to Enigma 1746, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1740 Sudoprime: The shaded numbers are 41 and 617

The winner Len Nunn of Bexley, Kent, UK

What are my two perfect squares?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 27 March. The Editor鈥檚 decision is final. Please send entries to Enigma 1738, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1732 Cache up front: 960 miles

The winner Mike Waterman of Camberley, Surrey, UK

Find the set of perfect squares that between them use each of the digits 0 to 9 exactly twice and whose sum is as small as possible. Your squares must all be different, and 0 may not be a leading digit.

What is the sum of your set of squares?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 26 September. The Editor鈥檚 decision is final. Please send entries to Enigma 1712, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1706 Festival figures: My three numbers are 487, 503 and 961

The winner Laura Marsden of Rotherham, South Yorkshire, UK

From a point on one side of a rectangular sheet of paper I drew two straight lines, one of them to a point on one adjacent side and the other to a point on the other adjacent side. My sheet of paper was now divided into two triangles and a pentagon. The lengths of the sides of the triangles were all integers, the lengths of the sides of the pentagon were, in some order, five consecutive integers, each less than 50.

What were the dimensions of the sheet of paper?

WIN拢15 will be awarded to the sender of the first correct answer opened on Wednesday 29 August. The Editor鈥檚 decision is final. Please send entries to Enigma 1708, 快猫短视频, Lacon House, 84 Theobald鈥檚 Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1702 All the sixes: The sum is 818181

The winner Graham Holmes of Ipswich, Suffolk, UK