Each of the 12 white squares in the cross-figure grid shown is to be filled with a single digit. The four two-digit numbers and the four three-digit numbers created are either perfect squares or perfect cubes. These squares and cubes are all distinct, and one of the down answers is both a square and a cube. What are the answers to 6 across and 7 across?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 28 August. The Editor’s decision is final. Please send entries to Enigma 1760, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1754 Elementary: There are 18 letters

The winner Andrew Cudbertson of Gorsedd, Flintshire, UK

The grid shows a cross-figure with two digits given as a start. All 11 two- and three-digit answers are prime and none has a leading zero; no digit appears more than once in any row, column or either of the two long diagonals; no even digit appears more than once anywhere.

What are the numbers in the shaded regions?

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WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 24 July. The Editor’s decision is final. Please send entries to Enigma 1755, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1749 Genotype: GENETICS is 25950365

The winner Len Nunn of Bexley, Kent, UK

In a new design of cellphone, each of the 10 number keys is associated with some letters of the alphabet. All letters each appear once, but are not in the usual alphabetical order found on a standard phone. For example, W appears on the zero button, S is on the 5 button and N is on the 9 button. It turns out that NEW=SCIENTIST÷ENIGMA.

What number is GENETICS?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 12 June. The Editor’s decision is final. Please send entries to Enigma 1749, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1743 Order, order!: The list of numbers is 8, 15, 19 and 29.

The winner Helen Wickins of Studley, Warwickshire, UK

In a new design of cellphone, each of the number keys 1 to 9 is associated with two or three letters of the alphabet. Predictive text software chooses letters for you as you type. The letters are not in the usual alphabetic order and there are no letters on the zero button. The eight words TWO, THREE,…. , NINE are each encoded by numbers that are divisible by the single digit implied by their names. Also, two consecutive items in that list are equal to their single digit raised to some power (a different power in each case). The word SCIENTIST is given by typing in a number that is a multiple of 11 and CLIVE is also a multiple of 11.

What number corresponds to CLIVE?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 8 May. The Editor’s decision is final. Please send entries to Enigma 1744, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1738 Two perfect squares: the squares are 1296 and 5625

The winner Anthony Buragina of Oakville, Ontario, Canada

The grid on the right shows a cross-figure with two digits given as a start. The 11 answers are distinct primes and none has a leading zero. No digit appears more than once in any row, column or either of the two long diagonals. What are the numbers in the shaded regions?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 10 April. The Editor’s decision is final. Please send entries to Enigma 1740, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1734 Friendly factors: PRINTED is 9478162

The winner Jason Dickson of Bretforton, Worcestershire, UK

In the country of Expluswye people do arithmetic using a smaller base than our decimal system, which uses base 10. My friend Basil recently moved house in Expluswye and, when he sent me his new address, commented that his new 3-digit house number was a perfect square. To my eyes Basil’s house number is a prime, but when I converted the number to decimal I saw that Basil was right and that the number is indeed a perfect square. This square written as a decimal is a 3-digit number that shares no digits with the house number as Basil writes it.

What is the number on Basil’s new house?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 20 March. The Editor’s decision is final. Please send entries to Enigma 1737, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1731 Power play: The difference is 7849

The winner Richard Brookfield of Worcester, UK

The grid shown on the right is a cross-figure puzzle with one digit given as a starting point. All 11 answers are prime numbers, none of them starts with a zero, and adding the individual figures in each column gives the same sum.

What are the two numbers in the shaded regions of the grid?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 30 January. The Editor’s decision is final. Please send entries to Enigma 1730, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1724 Bingo!: The top row numbers were: 2, 31, 43, 61 and 73

The winner Brian Knight of Cambridge, UK