Joe鈥檚 problem for Penny this week was to number each triangle of an icosahedron with one of the digits 1 to 5 so that each of the five triangles meeting at a point was numbered differently. Some of the digits in Penny鈥檚 solution are shown in the diagram of an icosahedron dissected and laid flat. Six of the digits are covered with the letters ENIGMA. What are those digits?

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 1780, 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 1774 March of the ants: The ants marched 11 centimetres

The winner Clive Stott of Bedford, UK

Penny knows how to construct a triangle when given the lengths of three sides, but doesn鈥檛 know how to construct a triangle given the heights of the triangle from two sides and the length of the third side. Joe suggested that with a ruler and a pair of scissors she might just find a way. As an example, Joe gave her the two heights of an acute-angled triangle as 5.6 and 6.0 centimetres and the length of its third side as 6.5 centimetres. How long were the other two sides?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 27 November. The Editor鈥檚 decision is final. Please send entries to Enigma 1773, 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 1767 Not over easy: Joe placed 36 cards sunny side up

The winner R. F. Tindall of Great Shelford, Cambridgeshire, UK

Joe always likes his fried eggs sunny side up. During breakfast in a motel in North Carolina he was given 36 cards, printed on one side with the motel鈥檚 name and address, and on the other with two eggs sunny side up. With the cards were instructions to place them on the table one by one and sunny side up to try to make a 6脳6 array with all cards sunny side up. As always with puzzles, there was a catch. If, as a card was placed, it touched edge to edge one or more cards, those cards had to be turned over. In solving the puzzle, Joe placed as many cards sunny side up as was possible.

How many did he manage?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 16 October. The Editor鈥檚 decision is final. Please send entries to Enigma 1767, 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 1761 Clock work: Esu rode the clock鈥檚 minute hand for 327 and 3/11 minutes

The winner Kevin Turner of Stoke-on-Trent, Staffordshire, UK

On the grid shown on the right, place the remaining 11 cards on the unshaded squares so that when you move from card to card the number of squares in the direction shown on each, you complete a circuit visiting all 12 unshaded squares once. What are the numbers on the cards, from top to bottom, in the third column?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 11 September. The Editor鈥檚 decision is final. Please send entries to Enigma 1762, 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 1756 Two triangles: The lengths of the sides of the triangles are 4, 3, 2 and 8, 7, 6.

The winner Martin Hooper of Burton upon Trent, Staffordshire, UK

In the 4脳4 grid shown, you must place all the unused arrow cards on the unshaded squares of the grid, such that by moving from card to card the number of squares in the direction shown on each, you complete a circuit visiting all 12 unshaded squares.

What are the numbers on the cards, from left to right, in the bottom row?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 5 June. The Editor鈥檚 decision is final. Please send entries to Enigma 1748, 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 1742 Chip-chop: The seven-figure number is 4625317

The winner Helen Gough of Southam, Gloucestershire, UK

Joe was having a dream. He was planning a solo trek to the North Pole and back, a total of 500 miles. His main problem was not being able to carry daily supplies sufficient for more than 240 miles, that is, in addition to the rest of his kit. He calculated how many supply dumps he would have to make on his way to the pole. After he had calculated the minimum total number of miles he would have to walk, setting up the dumps and getting to the pole and back, he woke up.

How many miles would Joe have to walk?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 13 February. The Editor鈥檚 decision is final. Please send entries to Enigma 1732, 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 1726 Conspicuous consumption: Fuel consumption for the return journey is 62 miles per gallon

The winner Tom Flannery of Derby, UK

Joe was in his school鈥檚 rugby team. While practising tackling, Joe remembers standing on the goal line 25 feet from his friend Ken, who was by the corner flag. When the master blew his whistle, Ken ran along the touchline and Joe had to try to tackle him. Ken runs at half a mile an hour faster than Joe, who could only manage 12 miles an hour. So Joe never caught up with Ken.

What was the nearest Joe came to Ken?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 16 January. The Editor鈥檚 decision is final. Please send entries to Enigma 1728, 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 1722 Double Latin: LATIN is 45123

The winner Ian McLaren of London, UK

Joe has a set of 25 cards, one side red and one side green. He has written one of the numbers 1 to 5 on each side of each card such that no two cards are identical. He asked Penny to arrange the cards in a 5-by-5 square, so that, red side up or green side up (after turning cards over one at a time), she produced a very special Latin Square in which not only were numbers not duplicated in any row, column or diagonal, but also where two cards touched corner to corner. The positions of some of the numbers in Penny鈥檚 solution are shown above.

What are the numbers hidden under L A T I N?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 5 December. The Editor鈥檚 decision is final. Please send entries to Enigma 1722, 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 1716 Pyramid selling: 3 tetrahedrons; 50 per cent increase

The winner Steve Collins of Reading, Berkshire, UK

Joe thinks he has found, by trial and error, a new property of triangles. He asked Penny to mark the points A along the sides of a triangle of her choice, each point to be a percentage P of the length of the side from the nearest corner. Then she had to join the points to form a second triangle and repeat the process to form a third triangle with points B. Penny found that the lengths of the sides of this third triangle were 52 per cent of the corresponding sides of her original. What was the value of P?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 7 November. The Editor鈥檚 decision is final. Please send entries to Enigma 1718, 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 1712 Each digit twice: The sum of the set of squares is 2439

The winner James R. Olson of Las Vegas, Nevada, US

Joe remembered a puzzle he solved years ago. In an alley, two ladders, each with their feet against the base of a wall, were leaning against the opposite wall. The distances (whole numbers of centimetres) above the ground where they touched the walls were given, and Joe had to calculate how high above the ground the ladders crossed. The answer was 105 centimetres. He also calculated that the ladders could have been 315 centimetres and 261 centimetres long. What was the width of the alley?

WIN 拢15 will be awarded to the sender of the first correct answer opened on Wednesday 5 September. The Editor鈥檚 decision is final. Please send entries to Enigma 1709, 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 1703 G&Ts all round: Amy was given a value of 36 for T

The winner John Shrimpton of Grizebeck, Cumbria, UK

When this Enigma was first posted, the sentence 鈥淭he answer was 105 centimetres鈥� was omitted by accident