A national charity is organising a tombola. It has printed tickets numbered 1, 2, 3,… and participants can pay a pound and choose one of the numbers at random. Just one per cent of the tickets earn the participant a prize.

In fact, a ticket is a winning one precisely when its number is a perfect power. So examples of winning numbers are:

1

25 (a square)

64 (a cube)

243 (a fifth power)

etc.

How many tickets were printed?

WIN £15 will be awarded to the sender of the first correct answer opened on Thursday 2 January. The Editor’s decision is final. Please send entries to Enigma 1777, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1771 Squares either way: 7 July 2056, 6 May 2016, 8 Jan 2016

The winner Michael Wickins of Studley, Warwickshire, UK

I have written down three 3-figure numbers which between them use nine different digits. Among these numbers there is: at least one that is divisible by 2; at least one that is divisible by 3; at least one that is divisible by 4; at least one that is divisible by 5; at least one that is divisible by 6; at least one that is divisible by 7; at least one that is divisible by 8; at least one that is divisible by 9.

Furthermore, all three numbers are divisible by 11. What are they?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 18 December. The Editor’s decision is final. Please send entries to Enigma 1776, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1770 Power point 2: 343, 243, 256, 216, 512

The winner Nick Smith of Chilcompton, Somerset, UK

Shown right is a cross-figure to complete. However, there are no across or down clues. Instead you have to ensure that: a) if an answer uses one of the numbered squares then that answer must be divisible by the little number in that square (for example the five-figure number in 1 down must be divisible by 1, 4 and 7; similarly the five-figure number in 7 across must be divisible by 7); b) in your answers, only two different digits can be used throughout, one of them odd and the other even; c) the odd digit must be written at most once in any row or column of the grid.

What are the answers to 1 across, and 7 across?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 20 November. The Editor’s decision is final. Please send entries to Enigma 1772, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1766 Triangular sums: The numbers are 6, 49, 23, 58 and 17

The winner Jesse Banwell of Ben Lomond, California, US

I wanted a cube-shaped dice with each of three of its faces having a perfect square number of spots and each of the remaining faces having a prime number of spots. To follow convention I wanted each face to have a different number of spots and I also wanted opposite pairs of faces to add up to the same total.

To make this dice I took eight identical standard dice and stuck them together to form a cube of double the size. On closer investigation I found that none of the faces of four spots from the eight original dice was visible on the exterior of my new dice.

How many faces of one spot and how many faces of two spots from the original dice were visible on the exterior of my new dice?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 23 October. The Editor’s decision is final. Please send entries to Enigma 1768, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1762 Quo vadis? II: The numbers from top to bottom in the third column are 3, 1, 2, 1

The winner Tony Harrison of Rugby, Warwickshire, UK

Eve said to me that she had in mind an even three-figure number that was divisible by 3. She also told me that she had spelled out the number in words and that she had counted the number of letters used. Knowing the number of letters would enable me to work out her number, she said.

Oddy said to me that he had in mind an odd three-figure number divisible by 3. He told me that he, too, had written the number in words and that he had counted the number of letters used. He said that knowing the number of letters would again enable me to work out his number.

Then the two of them had a little chat and announced that their numbers had no digit in common. What were their numbers?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 18 September. The Editor’s decision is final. Please send entries to Enigma 1763, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1757 Power point: The five numbers are 343, 243, 256, 216 and 512

The winner Alex Maynard of Ann Arbor, Michigan, US

Our local park is rectangular, with each of its sides a whole number of metres in length, the longer sides exceeding the shorter ones by 25 metres. The boundaries run north-south and east-west. There are two straight paths of equal length. One runs from the gate at the south-west corner of the park to a point on the northern boundary. The second is at right angles to the first and runs from a point on the first path to the gate at the south-east corner of the park. Each path is a whole number of metres long. How long are the paths?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 14 August. The Editor’s decision is final. Please send entries to Enigma 1758, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1752 Pentagon of squares: The smallest angle is 42 degrees; the angles either side are 64 and 121 degrees

The winner Trevor Morley of Weiterstadt, Germany

My textbook lists the two-letter symbols for chemical elements, including both old and new abbreviations for some of them, as: Ac Ag Al Am Ar As At Au Ba Be Bh Bi Bk Br Ca Cd Ce Cf Cl Cm Co Cr Cs Cu Db Ds Dy Er Es Eu Fe Fm Fr Ga Gd Ge Ha He Hf Hg Ho Hs In Ir Kr La Li Lr Lu Lw Md Mg Mn Mo Mt Na Nb Nd Ne Ni No Np Os Pa Pb Pd Pm Po Pr Pt Pu Ra Rb Re Rf Rg Rh Rn Ru Sb Sc Se Sg Si Sm Sn Sr Ta Tb Tc Te Th Ti Tl Tm Xe Yb Zn Zr.

Using letters no more than once, I have written as many as possible around a circle such that, if you look at any adjacent pair of letters, then reading them clockwise they are one of the above elements. How many letters have I written?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 17 July. The Editor’s decision is final. Please send entries to Enigma 1754, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1748 Quo Vadis?: The numbers on the cards in the bottom row of the grid are 1, 3, 2 and 1.

The winner Trevor Morley of Weiterstadt, Germany

I have drawn a circle, marked five points around its circumference, and joined each to the next by a straight line in order to make a pentagon. It turns out that the centre of the circle is outside this pentagon. I have then measured, in degrees, the five interior angles of the pentagon. The five numbers are all different and all but the smallest are perfect squares.

What is the smallest angle and what are the angles on either side of the smallest one?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 3 July. The Editor’s decision is final. Please send entries to Enigma 1752, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1746 Square in common: They both chose 3721

The winner Siu Loong Leong, New York, US

I applied for a job and had two interviews. The two interviewers had to decide independently whether or not I was suitable. Only after both interviews were the two decisions announced and the approval of both was needed to get the job. Apparently the first interviewer approved one in every P applicants (where P is a whole number) and decided that the rest were unsuitable. Then, for any of the first interviewer’s decisions, there was only a one in Q chance that the second interviewer agreed with the first (where Q is a larger integer). After the second interview, the interviewer told me that he had approved my application. However, he warned me not to get too excited because there was only a one in P+Q chance that I had got the job.

What are the values of P and Q?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 29 May. The Editor’s decision is final. Please send entries to Enigma 1747, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1741 Four squares: The five two-digit numbers are 56, 48, 72, 10 and 39

The winner Alan Walder of London, UK

I have written down a list of four positive whole numbers in increasing order. They are all less than 100, and no two of them have a common factor greater than 1. If I write the numbers in words, then each begins with a different letter of the alphabet and my list is also in alphabetical order. If I tripled the numbers, then the answers would be in reverse alphabetical order when written in words. If I then doubled those answers, the resulting numbers, when written in words, would be in alphabetical order once again.

What is my list of numbers?

WIN £15 will be awarded to the sender of the first correct answer opened on Wednesday 1 May. The Editor’s decision is final. Please send entries to Enigma 1743, ¿ìè¶ÌÊÓƵ, Lacon House, 84 Theobald’s Road, London WC1X 8NS, or to enigma@newscientist.com (please include your postal address).

Answer to 1737 Base jumping: The number on Basil’s new house is 647

The winner Peter Steinberg of Madison, Wisconsin, US