The letters S, N, A and P denote different digits, and SNAP is the corresponding four-figure number, which varies depending on choice of digits. It can be verified that, when SNAP is 1249, SNAP³ = 194844SNAP. What is the largest value SNAP can take and have SNAP³ ending with SNAP?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 13 March. The Editor’s decision is final. Please send entries to Enigma 1736, New ÐÓ°ÉÔ­´´, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1730 Prime position: The shaded numbers are 571 and 41

The winner Lizzie Dobson of Lancaster, UK

My uncle came to visit me by car on three days last week, taking the same round route each time, but returning by a different road from the one used to come to my hilltop home. On his arrival at my home on the first day, the miles per gallon (mpg) reading on his car’s display was 36.0, on the second 42.5 and on the third 44.0, when it also gave a reading of 71.5 for the cumulative mileage.

He told me that he had zeroed the display before he set out on day one, and had made no other journeys. He also says that, if I assume the mpg for the common approach and common return journeys are consistent from day to day, and that all the numerical values are exact, I should get a whole number answer for the mpg for the journey from my house back to his.

What is this whole number?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 2 January 2013. The Editor’s decision is final. Please send entries to Enigma 1726, New ÐÓ°ÉÔ­´´, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1720 Tables seating four: ENIGMA is 862735

The winner Nigel Brumpton of Newark, Nottinghamshire, UK

Russian peasants were said to perform multiplication knowing only how to add, and to multiply and divide by 2. I showed my niece how to multiply 25 by 31 by this method, as shown in the first box.

In the left column, the numbers are divided by 2 successively, any resulting halves being ignored, until 1 is reached. The right column consists of an equal number of values, each after the initial 31 being double the previous one. A simple rule, which I left my niece to discover, tells us which of the values in the right column we must add to get the required total of 775.

My brother-in-law then sent me the following problem, which uses the above method to multiply p by P, and which, with the relevant addition, gives the correct result, ENIGMA, where each of these six letters represents a different non-zero digit. He has inserted a 4 wherever it occurs in the calculation, leaving all other place-holders unspecified, as shown in the second box.

What is ENIGMA?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 21 November. The Editor’s decision is final. Please send entries to Enigma 1720, New ÐÓ°ÉÔ­´´, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1714 Penny-farthing: He was travelling at 28.8 km/hr

The winner David Barley of Gillingham, Kent, UK

I was staying at my sister’s house when my niece Amy came home from school feeling special. The class had been shown how to split a whole number, T, into whole number parts in such a way that the product of the parts was the greatest, G, that could be obtained for that T. For instance, she explained, 10 could be split into ten ones, or 2 and 4 and 4, or 5 and 5, and so on, which would yield products of 1, 32, and 25 respectively. But, she warned, G exceeds 32 for T=10.

Why did she feel special? Well, each pupil in the class had been given a different number in the range 20-50 inclusive for their personal T, and she had noted that, when she added the digits of her G together, the sum was exactly half of her T, and no one else in the class had T and G with this property.

What value of T was Amy given?

WIN £15 will be awarded to the sender of the first correct answer opened on Tuesday 19 August. The Editor’s decision is final. Please send entries to Enigma 1503, New ÐÓ°ÉÔ­´´, Lacon House,84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (include your postal address).

Answer to 1697 Binary palindrome: The number in decimal is 189

The winner Louella Harley of Belmont, Queensland, Australia

The number 3120 is divisible without remainder by each number in the following set: 3, 312, 12, 120, 2 and 20. Each of these numbers will be called a visible proper divisor (VPD) of 3120 because each is visible either as a single digit or as a group of adjacent digits in 3120. (1 and 3120 have been excluded as being improper, in order that any prime number has no VPDs).

ENIGMA represents an odd six-digit number in which all the digits are different. The set of all VPDs of ENIGMA is E, NI, IG, G, GMA, M, MA, and A. What is ENIGMA?

WIN £15 will be awarded to the sender of the first correct answer opened on Tuesday 1 May. The Editor’s decision is final. Please send entries to Enigma 1691, New ÐÓ°ÉÔ­´´, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1685 A noted eccentric: One or other of the clocks must have been read on a Tuesday

The winner Richard Crews of San Jose, California, US

My whimsical uncle tells me that he goes to his study at a certain time every day, but doesn’t say when. On Sundays when he arrives, he winds each of his two eight-day clocks and sets them to the correct time. One of the clocks gains an exact number of minutes each day, and the other loses the same, so the mean of the times is correct through the week.

He also tells me that on three days during a previous week (Sundays being excluded), as he entered the study he jotted down the time from one or other of the clocks, and he now produces the resulting slips of paper with the times 8:56, 9:11 and 9:20 written on them. These times are not necessarily in the order in which they were written down, nor do we know which clock is associated with which timing. Nevertheless, my uncle tells me I can deduce that he noted the time on a particular day of the week.

Which day was that?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 21 March. The Editor’s decision is final. Please send entries to Enigma 1685, New ÐÓ°ÉÔ­´´, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1677 The answer and winner will be announced next week

My niece, Angela, has two younger twin siblings, Ann and Ian. To celebrate the twins’ birthday, she asked me to compose a puzzle in which their names feature. Here it is:

With different letters denoting different digits consistently throughout, and with no leading zeros appearing, the number denoted by GEMINI is exactly divisible by the numbers corresponding to ANN and IAN. These two three-digit numbers have no common factor. The number denoted by ANGIE has digits which increase from left to right.

What number corresponds to ENIGMA?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 22 February. The Editor’s decision is final. Please send entries to Enigma 1681, New ÐÓ°ÉÔ­´´, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1674 Four months: FEB is 728

The winner John Lutz of Denistone, New South Wales, Australia