I have in mind three numbers. Each is a multiple of 3 and consists of three different non-zero digits, just one of which is 3. For each of these numbers individually, three of the following six statements are true and three are false. 1) It is the product of three different numbers, each a prime. 2) It is a triangular number, that is of the series 1,3,6,10,15 … 3) It is a cube, or a cube plus 3. 4) It may be written as a single-digit prime followed by a two-digit prime. 5) The 3 is the first or last digit. 6) It may be written as a two-digit prime followed by a single-digit prime.

What are my three numbers?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 11 December. The Editor’s decision is final. Please send entries to Enigma 1775, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1769 Crossing lines: I drew 15 lines

The winner Stephen Pollaine of Middletown, California, US

I have written a list of five different three-figure numbers, each of which is a power of a single digit. The first number is odd and thereafter each number has the same hundreds digit or the same tens digit or the same units digit as its predecessor. What (in order) are the five numbers?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 6 November. The Editor’s decision is final. Please send entries to Enigma 1770, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1764 Secret passages: Kathryn’s name in code is PZOSIBU

The winner Andy Lewis of Bourton-on-the-Water, Gloucestershire, UK

KATHRYN and her school friends have been using a Lorenz-type code to pass covert messages to each other. Each letter is expressed as a five-digit binary number such that A = 1 = 00001, M = 13 = 01101 and so on, but other symbols are represented by 00000 and by 11011 upwards. A fixed letter, say M, is chosen as a “coder”, known only to the sender and receiver. To transmit a letter, say D, it is added to the coder by the “exclusive-NOR” rule 1 + 1 = 1, 1 + 0 = 0, 0 + 1 = 0, 0 + 0 = 1. So, for example, D + M = 00100 + 01101 = 10110 = V. When the sent letter V is added by the recipient to the coder M, the original letter reappears: 10110 + 01101 = 00100.

She has sent her name to her friends as seven letters. KATHRYN and its coded version together consist of 14 different letters, so what was the coded version?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 25 September. The Editor’s decision is final. Please send entries to Enigma 1764, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1758 Path-o-logical: the paths are 156 metres long

The winner Helen Wickins of Studley, Warwickshire, UK

I have written a list of five different three-figure numbers, each of which is a power of a single digit. The first number is odd, and thereafter each number has the same hundreds digit or the same tens digit or the same units digit as its predecessor. What, in order, are the five numbers?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 7 August. The Editor’s decision is final. Please send entries to Enigma 1757, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1751 Rainbow square: The letters are RYGBVO

The winner Gordon Walsh of York, UK

My granddaughter has been given a set of 36 square tiles, six of each of the colours red (R), orange (O), yellow (Y), green (G), blue (B) and violet (V). In playing with these, she has arranged them into a square pattern in which each row, column and main diagonal contains each colour just once.

The colours in the top row are in the above order, left to right, while another row has the colours in the reverse order. One of the main diagonals contains consecutively in order the four tiles red, orange, yellow and green, or the reverse. From top to bottom what are the letters forming the other diagonal?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 26 June. The Editor’s decision is final. Please send entries to Enigma 1751, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1745 Cutting cubes: I started with four cubes

The winner Helen Gough of Southam, Gloucestershire, UK

I have a seven-figure number that uses seven consecutive digits in some order. Starting at the left and deleting a digit leaves a six-digit number, then deleting the right-end digit leaves a five-digit number, then deleting the left-hand end digit leaves a four-digit number, and so on, alternating sides until a single digit is left. Looking at the list of seven numbers obtained in this way I see that they are all odd, and that only the six-digit number is divisible by 3 (but not 9). Surprisingly, if I had carried out the process starting by deleting the right-hand end digit and then the left, and so on down to a single digit, all the above facts would still be true.

What number did I start with?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 24 April. The Editor’s decision is final. Please send entries to Enigma 1742, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1736 Child’s play: The largest value of SNAP is 9376

The winner Philip Belben of Coalville, Leicestershire, UK

I was talking to my granddaughter about the solar system and how planets and comets move in ellipses with the sun at one focus. I told her how you could draw an ellipse by sticking two pins in a sheet of paper both equidistant from and in line with the central point, placing a loop of string around them, and then by putting a pencil in the loop and keeping it tight, go right round. The pencil would thus draw the ellipse, each pin being at a focus.

I then asked her to draw the biggest ellipse she could on a sheet of paper 50 centimetres long and 30 cm wide, and then to find the area of the rectangle she could inscribe in her ellipse whose longest side was equal in length to that of the shorter side of the original sheet.

How far apart must she place the pins? How long was the string? What was the area of the inscribed rectangle?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 20 February. The Editor’s decision is final. Please send entries to Enigma 1733, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1727 Common factors: The consecutive numbers are 17, 18 and 19

The winner Ivan Simmons of Edinburgh, UK

The picture shows a typical card as used in our bingo club. Each column contains one, two or three numbers (at least one of each) increasing from top to bottom. Each row contains five numbers and four spaces. Column 1 has numbers from 1 to 9, column 2 from 10 to 19 and so on, with the last column from 80 to 89. On a card I had last night, in the top row all entries were primes and their sum was divisible by 2, 3 and 5; in the middle row all entries (and their sum) were 3 Ă— a prime; and in the bottom row all entries (and their sum) were 5 Ă— a prime. What were the five numbers in the top row?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 19 December. The Editor’s decision is final. Please send entries to Enigma 1724, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1718 Joe’s new property: 20 per cent

The winner Clive Bennett of Chichester, West Sussex, UK

Clearing my attic, I found my old 8mm cine-projector and films of our children. It still worked. The film moves through the projecting gate at an essentially constant speed from the stock spool to an identical take-up spool. The empty spool has a 4-centimetre diameter and the spool when full has a diameter (to the outside of the film) of 12 cm. The playing time is 17 minutes. Each spool has five spokes, so it is easy to compare speeds of rotation. A scene of children playing on the beach started when the ratio of the speed of rotation of the take-up spool to that of the stock spool was 5:3 and ended when it was 3:5.

How long did the scene last?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 31 October. The Editor’s decision is final. Please send entries to Enigma 1717, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1711 Illuminated artwork: A cycle takes 4 days and 15 minutes

The winner David Barley of Gillingham, Kent, UK

The penny-farthing was a precursor to the modern bicycle. It had one very large wheel with pedals, over which was the saddle, and one much smaller trailing wheel to enable steering. We have some very early film of my great-grandfather riding one he had built. It had a large wheel 6 metres in circumference with 48 spokes, and a small one of one-third that diameter having 20 spokes. It had no brakes. He was hurtling downhill -luckily at less than 50 km/h – and just before he fell off, the film appears to show both wheels not rotating. If the film is showing 16 frames per second how fast, in kilometres per hour, was he then travelling?

WIN ÂŁ15 will be awarded to the sender of the first correct answer opened on Wednesday 10 October. The Editor’s decision is final. Please send entries to Enigma 1714, żěè¶ĚĘÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1708 Pentagon: paper dimensions are 37 by 66

The winner Robert Armstrong of Thousand Oaks, California, US