A vertical piece of string is attached to a flat horizontal sheet of chicken wire, which forms a lattice of regular hexagons with 1-centimetre-long sides. The string is attached at one of the joints between three hexagons. Six ants marched down the string and along the wire, always moving further from their starting point on the wire. After a while, they had all marched the same distance along the wire (which was a whole number of centimetres less than 20 centimetres), but they were all at different straight-line distances from the starting point.

If I told you one of these straight-line distances, you would be able to calculate the straight-line distances the other five ants were from the starting point. How far had the ants marched along the wire?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 4 December. The Editor鈥檚 decision is final. Please send entries to Enigma 1774, 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 1768 Die hard: 8 ones and 4 twos were visible

The winner John Kelly of Walsall, West Midlands, UK

I have drawn a number of straight lines across a large sheet of paper, each extending from edge to edge on the paper, so that each line crosses all the other lines. One of the intersections is between three lines, all the others are between just two lines, and none of them are on the edge of the paper. I counted the number of non-overlapping areas formed that did not touch the edge of the paper and found that this was exactly three times the number of non-overlapping areas that did touch the edge of the paper. How many lines did I draw?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 30 October. The Editor鈥檚 decision is final. Please send entries to Enigma 1769, 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 1763 Clever spells: Eve and Oddy鈥檚 numbers are 378 and 201

The winner Bill Payne of Woking, Surrey, UK

I have before me some positive whole numbers, each consisting of a single digit, which may be repeated. The digit is different for each number, and the number of times it is repeated is also different for each number.

The sum of my numbers is a number in which each digit is larger than the digit on its left, and it is the largest number for which this is possible, given the constraints described above.

What is the sum of my numbers?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 2 October. The Editor鈥檚 decision is final. Please send entries to Enigma 1765, 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 1759 Cell count: the number of cells is 180

The winner Gordon Robson of Stockport, Cheshire, UK

Using the number grid shown (A) it is possible to generate nine-digit numbers in the following way: Start on any square and then move horizontally, vertically or diagonally to an adjacent square that hasn鈥檛 already been visited. Repeat until all nine squares have been visited. For example, the path shown in diagram B generates the number 235968741.

I have listed all the numbers that can be generated in this way, and that end in a certain digit. Some of these are divisible by four (and only four) of the numbers 1 to 9. The rest are divisible by five (and only five) of the numbers 1 to 9.

How many numbers are there in my list?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 19 June. The Editor鈥檚 decision is final. Please send entries to Enigma 1750, 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 1744 Clive鈥檚 number: CLIVE is 37125

The winner Simon Armstrong of Guisborough, Redcar and Cleveland, UK

I have before me a number of solid cubes. I make a single straight cut through each of them, avoiding cutting through any of the vertices. The resulting solids have between them the same number of even-sided faces as odd-sided faces. The number of cubes I started with is the minimum compatible with the information given above.

How many cubes did I start with?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 15 May. The Editor鈥檚 decision is final. Please send entries to Enigma 1745, 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 1739 Dodecagarden: The larger triangle was 147 times the area of the smaller

The winner John Crook of Betio, Republic of Kiribati

I have before me five two-digit numbers, with no leading zeros. All the digits are different and none of the numbers is prime. The sum of the five numbers is a perfect square. If I remove one of the numbers, the sum of the remaining four is also a perfect square. If I remove another number, the sum of the remaining three is again a perfect square, and if I remove a third number, the sum of the last two is again a perfect square.

What are my five numbers?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 17 April. The Editor鈥檚 decision is final. Please send entries to Enigma 1741, 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 1735 A pile of coloured cubes: 24 cubes are in the leftover pile

The winner Bernard Ambrose of Little Melton, Norfolk, UK

I have several boxes, each containing a number of cubes. Each cube has black and white faces (at least one of each colour per cube), and each box contains all possible different cubes, with no duplicates. My three nephews opened the first box and each tried to assemble their own 2脳2脳2 cube with only one colour on its outer faces, and of course they failed. They then opened further boxes until they were each able to assemble a single-coloured 2脳2脳2 cube. They then put all the leftover cubes from the open boxes in a pile.

How many cubes were there in the leftover pile?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 6 March. The Editor鈥檚 decision is final. Please send entries to Enigma 1735, 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 1729 Xmas gifts: A, N, S, W, E and R are 17, 4, 3, 26, 5 and 9

The winner Christopher R. Jeggo of Woking, Surrey, UK

I have in mind three consecutive two-figure numbers and I have calculated their sum and their product. I have then listed those positive integers which divide exactly into both that sum and product. It turns out that there are just six numbers in that list and that the sum of the six numbers is odd.

What are the three consecutive two-figure numbers?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 9 January 2013. The Editor鈥檚 decision is final. Please send entries to Enigma 1727, 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 1721 Odds and evens: The numbers being multiplied are 436 and 725

The winner Geoff Stone of North Bayswater, Victoria, Australia

I have before me a pile of nine counters numbered one to nine. I take two counters from the pile and write down the average of the numbers made by arranging them in two ways, that is in their original order and reverse order.

I then take a third counter from the pile and calculate the average of the numbers made by arranging the three counters in all possible orders. I repeat this until I have used all nine counters, ending up with eight averages. All but one of these averages are whole numbers, and the one that isn鈥檛 can be accurately written to one decimal place.

How many counters were used to generate the average that isn鈥檛 a whole number?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 14 November. The Editor鈥檚 decision is final. Please send entries to Enigma 1719, 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 1713 Inventories: The number is 3223

The winner Heather Myers of St John鈥檚, Newfoundland, Canada

In our town square is a piece of installation art consisting of 300 rods of equal length arranged in a cubic lattice. The whole structure balances on a single vertex, with the diagonally opposite vertex at the top. Each rod contains an LED strip light, which can be independently switched on and off. A microprocessor controls this, lighting up the rods that form the shortest continuous path between the bottom and top vertices.

Each combination of rods (meeting the shortest continuous path criterion) is lit up for 10 seconds, then another and so on, until all combinations have been displayed, when the cycle starts again.

How long does it take to complete a full cycle?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 19 September. The Editor鈥檚 decision is final. Please send entries to Enigma 1711, 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 1705 Not deducible: Harry鈥檚 triangular number is 378 and Tom鈥檚 perfect square is 6084

The winner Frank McCarthy of Liverpool, UK