Enigmatic Code

Programming Enigma Puzzles

Category Archives: enigma

Enigma 419: Painting by numbers

From New Scientist #1569, 16th July 1987 [link]


1. You will need four copies of:

Enigma 419 - 1

labelled A, B, C, D.

2. Take A. The artist Pussicato signs the top row and you sign the bottom row; your signature must contain 9 letters.

3. Fill in B by using A as follows. Take each A square in turn, find the position of its letter in the alphabet and from that number subtract the appropriate multiple of 5 to leave a number from 1 to 5. Put that number in the corresponding B square.

4. Fill in C by using B as follows. Each square touches three or five other squares, including touching along a side or at a corner. Take each B square in turn and add up the numbers in the squares it touches. From the total subtract the appropriate multiple of 5 to leave a number from 1 to 5. Put that number in the corresponding C square.

5. Paint D by using C as follows.Number the five colours, Red, Blue, Green, Yellow, White, 1 to 5 in any order you like. Take each C square in turn and find the colour you have given its number. Paint the corresponding D square with that colour.


Enigma 419 - 2

A painter whose name involves only the first five letters of the alphabet produced:

Enigma 419 - 3

What was the painters name?



Enigma 1092: A prime age

From New Scientist #2248, 22nd July 2000

Marge, April, May, June, Julia and Augusta have all celebrated their birthday today. They are all teenagers and with the exception of the one pair of twins their ages are all different.

Today, only Marge and April have ages which are prime numbers, but the sum of the ages of all the girls is also a prime number. On their birthday last year, only May and June had ages which were prime numbers, but again the sum of the ages of all the girls was a prime number. On their birthday two years before that, only May and Julia had ages which were prime numbers, but even then, the sum of the ages of all the girls was again a prime number.

How old is Augusta?


Enigma 418: Let us divide

From New Scientist #1568, 9th July 1987 [link]

In the following division sum each letter stands for a different digit:

Re-write the sum with the letters replaced by digits.


Enigma 1093: Primed to spend

From New Scientist #2249, 29th July 2000

Bill’s credit card has the usual four four-digit numbers, which are in ascending order of size. All are prime numbers and the sum of the digits of each is the same.

The digits in the first number are all different and the third number is the first number reversed. The digits in the second number are all different and the fourth number is the second number reversed. The last digits of the four numbers are all different.

He has a hopeless memory for figures, but he can always work out his four-digit PIN from his card, because he can remember that it is equal to the difference between the first and third numbers (or the difference between the second and fourth numbers, which is the same) and happens to be a perfect square.

What is the fourth number?


Enigma 417: Snooker triangle

From New Scientist #1567, 2nd July 1987 [link]

We have a small snooker table at home, everything being in a reduced form of the real thing. The point system is the same: that is, 1 for a red, with each potted red enabling the player to try for one of six colours with points from 2 to 7. (For example, the blue is worth 5). At the end of the frame the six colours are potted in turn.

Last Saturday, I played with my daughter and the frame was completed in just one visit to the table by each of us. I opened, potted a red with my first shot, then potted a colour (which, of course, was brought out again) and then, whenever I successfully potted a colour after a red, it was always that same colour. Then I made a mistake (without any penalties) and my daughter took over. She, too, always followed a red by a particular colour, but a different one from mine. She cleared the table and we drew on points: we decided to replay the next day.

Surprisingly, all that I said about Saturday’s frame could be said about Sunday’s, but this time we drew with one more point each than on the previous day.

How many times, in total for the two frames, did I pot the blue?

How many balls does my small table have?


Enigma 1094: De-fence

From New Scientist #2250, 5th August 2000

In my garden there is a circular pond less than two metres across. Because my young nephew was coming to stay I asked a local handyman to erect a fence around it. He did this by taking three straight lengths of fencing, two of them equal, and each of them a whole number of metres long. He formed these into a triangle which fitted around the pond.

I complained that this took up too much space, so he adapted the construction to make a hexagonal fence around the pond. Opposite sides of the hexagon were parallel, three of the sides used bits of the original triangular fence without moving them, and all six sides touched the edge of the pond. The total perimeter of the new hexagonal fence was precisely half of that of the original triangular fence.

What were the lengths of the three original straight pieces of fencing?


Enigma 416: Short notice

From New Scientist #1566, 25th June 1987 [link]

I am making a noticeboard from a sheet of cork. In my search for some wood to back it, I came across two pieces having the same area as the cork, but different dimensions. To make the first piece fit, it would have been necessary to cut A feet off the cork and B feet off the wood, and to fit the second would have required C feet to be cut off the cork and D feet off the wood.

When I arrived at the timber yard I had forgotten the dimensions of my piece of cork. I remember only that, in measuring A, B, C and D, I obtained 1 foot, 2 feet and 4 feet only, with one measurement occurring twice. Which one occurred twice and in what order I obtained these measurements I forget.

All I can remember is that the cork measured a whole number of inches along each side and that none of its sides measured a whole number of feet.

Can you help me to deduce the size of wood I need to buy before the woodyard closes?

(Answers in inches, please: 1 foot = 12 inches).


Enigma 1096: Prime break

From New Scientist #2252, 19th August 2000

At snooker a player scores 1 point for potting one of the 15 red balls, but scores better for potting any of the 6 coloured balls: 2 points for yellow, 3 for green, 4 for brown, 5 for blue, 6 for pink, 7 for black.

Davies potted his first red ball, followed by his first coloured ball, then his second red ball followed by his second coloured ball, and so on until he had potted all 15 red balls, each followed by a coloured ball. Since the coloured balls are at this stage always put back on the table after being potted, it is possible to pot the same coloured ball repeatedly.

Davies’ break was interesting as after he had potted each of the 15 coloured balls his cumulative score called by the referee was always a prime number.

After potting the 15 red balls and 15 coloured balls, a player’s final task is to attempt to pot (in this order) yellow, green, brown, blue, pink and black. I won’t tell you how many of those Davies managed to pot, nor could you be sure how many of them he potted even if I told you his total score for the break.

What was that total score?


Enigma 415: Buses galore

From New Scientist #1565, 18th June 1987 [link]

The Service Bus Company runs buses on the route shown by the map:

Enigma 415

Each bus starts its journey at the Terminus, T, and goes towards A. At each crossroads it goes straight across. The bus eventually enters T from B and leaves for C. It finally arrives at T from D, to complete its journey. The time to go from one crossroads to the next is three minutes and so it takes one hour to complete the journey.

Buses are timetabled to leave T on the hour and at various multiples of three minutes after the hour. When a bus completes a journey it immediately begins its next journey and so the timetable repeats each hour.

This means each bus reaches a crossroads at times 00, 03, 06, 09, …, 54, 57. The buses are timetabled so that no two buses ever reach the same crossroads at the same time. Also the buses are timetabled so that the maximum number of buses are running on the route.

How many buses are running on the route?


Enigma 1097: Chessboard triangles

From New Scientist #2253, 26th August 2000

Take a square sheet of paper of side 1 kilometre and divide it into small squares of side 1 centimetre. Colour the small squares so as to give a chessboard pattern of black and white squares.

When we refer to a triangle, we mean a triangle OAB, where O is the bottom left corner of the square of paper, A is on the bottom edge of the paper and B is on the left hand edge of the paper.

Whenever we draw a triangle then we can measure how much of its area is black and how much is white. The score of our triangle is the difference between the black and white areas, in square centimetres. For example if OA = 3 cm and OB = 2 cm then we find the score of the triangle is 1/6 cm².

Question 1. What is the score of the triangle with OA = 87,654 cm and OB = 45,678 cm?

Question 2. What is the score of the triangle with OA = 97,531 cm and OB = 13,579 cm?

Question 3. Is it possible to draw a triangle on the paper with a score greater than 16,666 cm²?


Enigma 414: Nicely bungled, Sir!

From New Scientist #1564, 11th June 1987 [link]

My Uncle, I fear, has done it again. He has been taking great interest in the activities of four local football teams — A, B, C and D — and he managed to obtain some details, not very complete ones, I am afraid, of the matches played, won, lost, drawn and so on. But no one will be surprised, perhaps not even Uncle Bungle, to hear that one of the figures given was incorrect.

The details that he had looked like this:

Enigma 414

Which figure was wrong? What was the score in the matches that had been played?


Enigma 1098: Soccer heroes

From New Scientist #2254, 2nd September 2000

There are seven teams in our local football league. Each team plays each of the others once during the season. We are approaching the end of the season and I have constructed a table of the situation so far, with the teams in alphabetical order.

Here are the first two rows of the table, but with digits consistently replaced by letters, different letters being used for different digits.

What was the score when Albion played Borough?


Enigma 413: Quargerly dues

From New Scientist #1563, 4th June 1987 [link]

A native of Kipwarm had a gold necklace consisting of links joined together to form one long unbroken loop of chain.

He has fallen on hard times and to pay his gas bill he is going to give the gas board one link of his gold necklace every day.

He has broken just a certain number of links (thus forming that number of individual links and some other variously-sized pieces of chain). By giving away and taking back certain pieces he can ensure that, at the rate of one a day, his total of links decreases and the board’s increases. Furthermore, had his necklace had one more link, it would have been necessary to break one more in order to pay the board in this way.

His necklace will last him a whole number of quargers (a Kipwarmian period of a certain number of days, less than one year). But the board has offered him an alternative way of paying. His first quarger’s gas will be free, the next will cost him one link, the next two links, the next quarger’s will cost him four links, and so on, doubling each quarger. At that rate the necklace will pay for the same number of days’ gas.

How many days are there in a Kipwarmian quarger?


Enigma 1099: Unconnected cubes

From New Scientist #2255, 9th September 2000 [link]

I have constructed a cyclical chain of four-digit perfect cubes such that each cube in the chain has no digits in common with either of its neighbours in the chain. The chain consists of as many different four-digit cubes as is possible, consistent with the stipulation that no cube appears in it more than once.

If I were to tell you how many cubes lie between 1000 and 1331 either by the shorter route or by the longer route round the chain you could deduce with certainty the complete order of the cubes in the chain.

Taking the longer route round the chain from 1000 to 1331, list in order the cubes that you meet (excluding 1000 and 1331).


Enigma 412: A triangular square

From New Scientist #1562, 28th May 1987 [link]

Enigma 412

Professor Kugelbaum, deep in thought and in a distracted state, wandered onto a building site. He saw a man laying equilateral triangular slabs on a plain flat area. Turning his keen mind from the abstract to the concrete, he asked the man with a sudden inspiration, “What are you doing?”

“I’m laying a town square.”

“But the angles aren’t right.”

“Well, it’s going to be a square in the form of an enormous equilateral triangle”, was the reply.

“I don’t call nine slabs enormous.”

“Ah”, said the workman, “first, I haven’t finished yet: I’ve just started at one apex. Secondly, if you look carefully, you’ll see that there are in fact 13 different triangles to be found in the pattern I’ve already laid [see diagram]. When I’ve finished there will be 6000 times as many more triangles to be found in the completed array.”

Kugelbaum’s mind began to tick over.

How many slabs will there be in the completed array?

This puzzle brings the total number of Enigma puzzles on the site to 1,100 (and by a curious co-incidence on Monday I posted Enigma 1100 to the site). This means there are (only!) 692 Enigma puzzles remaining to post, mostly from the 1990s. There is a full archive of puzzles from the inception of Enigma in February 1979 up to May 1987 (this puzzle), and also from September 2000 up to the end of Enigma in December 2013. Happy puzzling!


Enigma 1100: Sydney 2000

From New Scientist #2256, 16th September 2000 [link]

Whether it’s the “Sydney 2000 games in year 2000” or the “Games in year 2000 in Sydney”, either way, both are arranged in additions (I) and (II), where the only given digits appear in the number 2000 as shown and all other digits have been replaced by letters and asterisks.

In these additions, different letters stand for different digits and the same letter always stands for the same digit whenever it appears, while an asterisk can be any digit.

What is the numeric value of SYDNEY?


Enigma 411: The third woman

From New Scientist #1561, 21st May 1987 [link]

The Ruritanian Secret Service has nine women agents in Britain: Anne, Barbara, Cath, Diana, Elizabeth, Felicity, Gemma, Helen and Irene. Any two of the women may or may not be in contact with each other.

To preserve security contacts are limited by the following rule: for any two of the women there is a unique third woman who is in contact with both of the women. The British Secret Service has so far discovered following pairs of women that are in contact: Anne and Cath, Anne and Diana, Cath and Barbara, Barbara and Gemma, Elizabeth and Felicity.

Which of the women are in contact with Helen? Who is the woman in contact with both Anne and Irene?


Enigma 1101: Disappearing numbers

From New Scientist #2257, 23rd September 2000 [link]

This game starts when I give a row of numbers; some numbers in italic [red] and some in bold [green]. Your task is to make a series of changes to the row, with the aim of reducing it to a single number or to nothing at all. In each change you make, you select two numbers that are adjacent in the row and are of different font [colour], that is to say one is italic [red] and the other is bold [green]. If the numbers are equal, you delete them both from the row; otherwise you replace them by their difference in the font [colour] of the larger number.

For example, suppose I gave you the row:

3, 4, 3, 2, 5, 2.

One possibility is for you to go:

[I have indicated the pair of numbers that are selected at each stage by placing them in braces, the combined value (if any) is given on the line below in square brackets].

3, 4, 3, {2, 5}, 2
3, {4, 3}, [3], 2
{3, [1]}, 3, 2
[2], {3, 2}
2, [1]

You have come to a halt and failed in your task.

On the other hand you could go:

{3, 4}, 3, 2, 5, 2
[1], {3, 2}, 5, 2
{1, [1]}, 5, 2
{5, 2}

And you have succeeded in your task.

For which of the following can you succeed in your task?

Row A: 9, 4, 1, 4, 1, 7, 1, 3, 5, 4, 2, 6, 1, 4, 8, 3, 2.

Row B: 2, 3, 5, 9, 6, 3, 1, 4, 2, 3, 1.

Row C: 1, 2, 3, 4, 5, 6, …, 997, 998, 999, 1000, 3, 5, 7, 9, 11, …, 993, 995, 997, 999.

Row D: 3, 2, 1, 4, 5, 4, 3, 2, 4, 3, 7, 4, 1, 5, 1, 4, 2, 4, 3, 1, 2, 7, 9, 3, 7, 5, 3, 8, 6, 5, 8, 4, 1, 5, 2, 3, 1, 4, 10, 6, 3, 5, 7, 4, 1, 4.

I have coloured the numbers in italics red, and those in bold green in an attempt to ensure the different styles of numbers can be differentiated.

When the problem was originally published there was a problem with the typesetting and the following correction was published with Enigma 1104:

Due to a typographical error, three of the numbers in Enigma 1101 “Disappearing Numbers” appear in the wrong font. In each case, the following should have been printed in heavy bold type:

the second number 3 in the initial example;
the first number 5 in row B;
and the first number 4 in the second line of row D.

I have made the corrections to the puzzle text above.


Enigma 410: Most right

From New Scientist #1560, 14th May 1987 [link]

The addition sums which Uncle Bungle has been making up recently, with letters substituted for digits, have been getting longer and more complicated. And no one will be surprised to hear that in the latest one everything is not as it should be. In fact one of the letters is wrong.

Here it is:

What can you say about the letter which is wrong? What should it be? Find the correct sum.


Enigma 1102: The Apathy Party

From New Scientist #2258, 30th September 2000 [link]

George called a meeting to inaugurate the National Apathy Party, open to anyone who has never voted in a General Election. He hopes to be the next Prime Minister. The turnout was phenomenal, but he managed to seat them all in Wembley Stadium (capacity 80,000). George proposed that the President and the Committee should be chosen by chance, rather than by ballot. The delegates had been allocated sequential membership numbers of arrival — George, of course, being No. 1. He proposed that one number be chosen at random by the computer — that member would be the President. All members whose numbers divide exactly into the President’s number would be on the Committee. Apathy reigned — this totally undemocratic procedure was agreed.

The computer produced an odd membership number for the President and the number of committee members, including George and the President, was an odd square greater than 10.

What was the President’s membership number?