Enigmatic Code

Programming Enigma Puzzles

Tag Archives: by: Keith Austin

Enigma 441: The coloured painting

From New Scientist #1591, 17th December 1987 [link]

I looked down at the body slumped over my desk. One hand held my card “Newton Harlowe — Private detective”, and the other a painting. All I knew about painting came from watching my secretary Velda doing her nails. However, I could see in the dim light that is was a 6 × 6 array of small squares, each coloured red or blue or green. As the neon lights on the nightclubs opposite my office window flashed on and off and the light reflected from the wet sidewalks, I was able to make out the vertical columns of the painting. I saw:

though that was not necessarily the order they occurred in the painting. Suddenly the door opened and a raincoated figure with an automatic entered. There was a loud bang and everything went black.

I came round to find myself lying next to the body of a blonde on the floor of a living room. From the sound of the surf outside I could tell it was a beach-house. There on the wall was the painting. The moonlight shone onto it through the shutters. As they moved in the breeze I was able to make out the horizontal rows of the painting. I saw:

though again not necessarily in the right order. Just then a police siren sounded outside. I was going to have to do some explaining, and that painting was the key.

Reproduce the painting.



Enigma 1070: Time to work

From New Scientist #2226, 19th February 2000

Amber cycles a distance of 8 miles to work each day, but she never leaves home before 0730h. She has found that if she sets off at x minutes before 0900h then the traffic is such that her average speed for the journey to work is (10 − x/10) miles per hour. On the other hand, if she sets off at x minutes after 0900h then her average speed is (10 + x/10) miles per hour.

(1) Find the time, to the nearest second, when Amber should set off in order to arrive at work at the earliest possible time.

Matthew lives in another town but he also cycles to work, setting off after 0730h, and he has found that his average speed for the journey to work follows exactly the same pattern as Amber’s. He has calculated that if he sets off at 0920h then he arrives at work earlier than if he sets off at any other time.

(2) How far does Matthew cycle to work?


Enigma 1072: Into three piles

From New Scientist #2228, 4th March 2000

Sunny Bay fisherfolk have a tradition that when they return home with a catch of fish they take all the catch and divide it into three piles. Over the years they have pondered the question: given a particular number of fish, how many different ways can they be divided up? For example, they could divide up 10 fish in 8 ways, namely, (1, 1, 8), (1, 2, 7), (1, 3, 6), (1, 4, 5), (2, 2, 6), (2, 3, 5), (2, 4, 4) and (3, 3, 4).

One day the fisherfolk netted four large sea shells. On one side of each was one of the letters A, B, C and D and each shell carried a different letter. Each shell also had on its reverse one of the numbers 0, 1, 2, 3, 4, 5, 6, … The fisherfolk found that if they caught N fish then the number of different ways of dividing them into three piles was:

[(A × N × N) + (B × N) + C] / D

rounded to the nearest whole number. (Whatever the number of fish, the calculation would never result in a whole number plus a half; so there was no ambiguity about which whole number was the nearest).

I recall that D was less than 21, that is, the number on the reverse of the shell with D on it was less than 21. Also A and C were different.

What were A, B, C and D?


Enigma 437: Find the fields

From New Scientist #1587, 19th November 1987 [link]

Long Acre Farm measures 6 furlongs by 10 furlongs as shown on the map; the dotted lines are at furlong intervals.

Enigma 437

In the old days, the farm was divided into 12 rectangular fields by straight hedges running north-south or east-west right across the farm. The dimensions of the fields were all whole numbers of furlongs.

Recently, five stones were discovered, bearing numbers and located as shown on the map. It appears that each such stone indicated the area, in square furlongs, of the field it was in.

Draw a map showing the 12 fields.


Enigma 432: Holiday on the islands

From New Scientist #1582, 15th October 1987 [link]

Alan and Susan recently spent eight days among the six Oa-Oa islands, which are shown on the map as Os.

Enigma 432

Only two of the islands, Moa-Moa and Noa-Noa, have names and hotels. The lines indicate the routes of the four arlines: Airways, Byair, Smoothflight and Transocean.

Alan and Susan started their holiday on the morning of the first day on Moa-Moa or Noa-Noa. On each of the eight days they would fly out to an unnamed island in the morning and then on to a named island in the afternoon and spend the night on that island. They each had eight airline tickets and each ticket was a single one-island-to-the next journey for two passengers. Alan had two Airways and six Byair tickets, while Susan had three Smoothflight and five Transocean tickets. They noticed that whatever island they were on, only one of them would have tickets for the flights out and so they agreed that, each time, that person should choose which airline to use.

Now Alan preferred that they should spend the nights on Moa-Moa, while Susan preferred Noa-Noa. However, they are an inseparable couple. So they each worked out the best strategy for the use of their tickets in order to spend the maximum number of nights on their favourite island.

How many nights did they spend on each island?


Enigma 1082: End-of-season blues

From New Scientist #2238, 13th May 2000 [link]

In 1998-99 our local hockey team had 5 teams, Ant, Bat, Cat, Dog and Emu. Each team played every other team once during the season. There were no drawn games as penalty shoot-outs were used when required. One point was awarded for a win. At the end of the season the teams were ordered in a table according to points and teams with equal points were bracketed together. The only game I saw was when Cat beat Dog. The final game of the season was Ant versus Bat. Unfortunately, before it was played it was clear that whatever the result of the game, it was going to have no effect on the order and bracketing of the teams in the end-of-season table.

Question 1. Who won the game between Cat and Emu?

In 1999-00 the number of teams went up to 7 with the addition of Fox and Gnu. The rules were not changed. I remember that Bat beat both Cat and Dog; however Bat finished the season with fewer points than Ant. The final game of the season was again Ant versus Bat and again it was clear beforehand that the result would have no effect on the order and bracketing of the table.

Question 2. How many points did Emu have at the end of the season and who won the games Cat v Fox and Dog v Gnu?

Happy Christmas from Enigmatic Code!


Enigma 428: The torn map

From New Scientist #1578, 17th September 1987 [link]

Nostalgia Pictures are remaking that old Saturday matinée movie serial The Torn Map. This tells how the heroine searches for the map which will reveal the whereabouts of the treasure she has inherited. The map has been torn into 12 parts, and she retrieves one part in each episode of the serial. Incidentally, you might recall the titles of those episodes: “Trail of the map”, “Flames of death”, “Avalanche horror”, “Beneath thundering hooves”, “Lift-shaft terror”, “Poison peril”, “Cliffs of doom”, “Snakes of menace”, “Plunge to disaster”, “Bombs of vengeance”, “Crash of danger”, “Secret of the map”.

In the final episode the heroine has the 12 numbered pieces on the table in front of her. To her horror, she finds that each piece had part of the real map on one side and part of a fake map on the other. Thus she will have to turn some of the pieces over so that all 12 real parts are face up. To help her, each piece has some letters on each side. For example, piece 5 has F on the face-up side, and I and E on the face-down side. If she can turn over some of the pieces so that every letter in the alphabet from A to O is showing face-up, then she will have the 12 real parts of the map showing.

The letters on the pieces are as follows, with letters on the face-up side given first, and then those on the face-down side:

(1) C/H;
(2) G/N;
(3) D/F;
(4) J/G;
(5) F/I,E;
(6) B/L,H;
(7) I/E,D;
(8) A/KO;
(9) B/M,E;
(10) O,N/K;
(11) M,L/C;
(12) B,N/A.

Which pieces should she turn over?


Enigma 1084: 1-2-3 triangles

From New Scientist #2240, 27th May 2000 [link]

The diagram shows a large triangle divided into 100 small triangles. There are 66 points that are corners of the small triangles.

You are to write 1 or 2 or 3 against each of the 66 corner points. The only restrictions are:

(a) the corners of the large triangle must be labelled 1, 2 and 3 in some order;
(b) each number on a side of the large triangle must be the same as the number at one end of that side.

Q1: Is it possible for you to write the numbers on so that there are precisely 10 small triangles with corners labelled 1, 2 and 3?

Q2: As Q1 but with 32 small triangles.

Q3: As Q1 but with 61 small triangles.

Q4: As Q1 but with 89 small triangles.


Enigma 424: A round of fractions

From New Scientist #1574, 20th August 1987 [link]

Anne and Barbara have just played a round of golf consisting of 18 holes. Anne decided to keep an unusual record of the game. After each hole she formed the fraction consisting of her total score to that point divided by Barbara’s. Naturally she reduced each fraction to its lowest terms. For example, if, after seven holes, Anne had had a total score of 33 strokes Barbara one of 30 strokes, then Anne would have recorded the fraction 11/10. The fractions Anne obtained are as follows, in increasing order, not necessarily in the order they occurred in the game:


On the course that Anne and Barbara played on, each hole was par 4, that is, at each hole a player is expected to take four strokes. In their round, each girl, at each hole, scored par or a birdie or a bogey. A birdie is a score one less than par and a bogey is a score one more than par.

Which holes did Anne win, that is, take fewer strokes at, which did Barbara win, and which were shared?


Enigma 1089: Catch the buses

From New Scientist #2245, 1st July 2000 [link]

The island of Buss is divided into 3 counties, A-shire, B-shire and C-shire, each containing a number of towns. Each town in A-shire has just one road and that goes to a town in B-shire. Each town in B-shire, irrespective of how many roads it has to towns in A-shire, has just one road that goes to a town in C-shire. There are 3 bus companies, Red, Yellow and Green. Here are the buses for today:

• From each A-shire town one Red bus runs to a B-shire town.
• From each B-shire town one Yellow bus runs to a C-shire town.
• From each A-shire town one Green bus runs to a C-shire town.

Naturally the destinations are determined by the roads. A bus company is “economical” if no town is the destination of more than one of its buses today. A bus company is “covering” if every town in the county its buses today finish in is the destination of at least one of those buses. Which of the following statements are bound to be true:

(1) If the Green Co is economical then the Red Co is economical.
(2) If the Green Co is not economical then the Red Co is not economical.
(3) If the Green Co is covering then the Yellow Co is covering.
(4) If the Green Co is not covering then the Red Co is not covering.
(5) If the Green Co is economical and the Red Co is covering then the Yellow Co is economical.
(6) If the Green Co is covering and the Yellow Co is economical then the Red Co is covering.

This is another puzzle from an old paper copy of New Scientist that I recently found. It currently doesn’t appear in the on-line archives.


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 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 [link]

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 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 1105: Road ants

From New Scientist #2261, 21st October 2000 [link]

Take a large sheet of paper and a black pen and draw a rectangle ABCD with AB = 10 metres and BC = 2 metres. Now draw lines to divide your rectangle into small squares, each of side 1 centimetre. Place your diagram so that A is due north of D and B is east of A. In each small square draw the diagonal that goes from northwest to southeast. Let P and Q be the mid-points of AD and BC, respectively. Then there is a black line PQ; remove it and replace it by a red line.

Amber is a small ant who can walk along the black lines in your diagram. North of PQ she covers a centimetre in 1 minute, but south of PQ she can cover a centimetre in 30 seconds. She is to walk from C to A and she chooses the quickest route.

1. How long does Amber take on her journey? Give the time, to the nearest second, in hours, minutes and seconds.

Ben is another ant who walks along the black lines. North of PQ he goes at the same speed as Amber, but not south of PQ. The fastest time for Ben to get from C to A is 24 hours.

2. South of PQ, how long does Ben take to cover a centimetre? Give the time, to the nearest second, in minutes and seconds.


Enigma 406: The ritual

From New Scientist #1556, 16th April 1987 [link]

I entered the jungle clearing and found, at the centre, six stones numbered 1, 2, 3, 4, 5, 6. My native guide demonstrated the ancient ritual which was enacted on that site.

First I had to arrange the stones in any order I wished, so I put them as 421365. I then noted the number on the first stone — all counting is from the left — is was 4, and so I picked up the fourth stone and placed it at the right hand end. I obtained 421653. Next, I looked at the second stone, a 2, and moved the second stone to the end, to give 416532. I repeated the procedure for the third stone to give 416532, and then for the fourth, fifth and sixth stones, to give successively, 416523. 465231, 652314. The final arrangement was called the result of the ritual.

Just then the high priestess, Sarannah, entered. She arranged the stones in a certain order, carried out the ritual and obtained the result 314625. My guide explained that the arrangement Sarannah started with was the result of applying the ritual to a very special arrangement, which she could not tell me.

What was the arrangement that Sarannah started with?


Enigma 402: A DIY puzzle

From New Scientist #1552, 19th March 1987 [link]

In the puzzle below, select some of the guest lists and discard the remainder. The lists you keep should make a puzzle which has exactly one solution and involves no more lists than is necessary.

Who did it?

by …

There has been a series of robberies at house parties recently. Each was clearly a one-man job — the same man each time — and each was an inside job. The possible suspects are Alan, Bryan, Chris, David, Eric, Fred, George, Harry, Ian, Jack, Ken and Len. The male guest list at the parties were as follows:

1. All but David, George and Len.
2. Bryan, Chris, Eric, Harry, Ian and Ken.
3. All but Bryan and Ken.
4. Chris and Ian.
5. All but Alan, Fred, Jack and Len.
6. Bryan, Chris, Ian and Ken.
7. All but Eric and Harry.
8. All but David and George.
9. All but Chris and Ian.
10. All but Alan and Fred.

Who carried out the robberies?

Which lists should you use in your puzzle?

What is the answer to your puzzle?


Enigma 1110: Dots and lines

From New Scientist #2266, 25th November 2000 [link]

Matthew and Ben are playing a game. The board is a 1-kilometre square divided into 1-centimetre squares. The centre of each small square is marked by a red dot.

Matthew begins the game by choosing a number. Ben then selects that number of red dots. Finally Matthew chooses two of Ben’s selected dots and draws a straight line from one to the other. Matthew wins if his line passes through a red dot other than those at its ends; otherwise Ben wins.

What is the smallest number that Matthew can choose to be certain of winning?

In the magazine this puzzle was incorrectly labelled Enigma 1104.


Enigma 398: Down on the farm

From New Scientist #1548, 19th February 1987 [link]

Farmer O. R. Midear has crossed a turnip with a mangel to get a tungel, and he now has many fields of tungels. As a tungel expert, he looks at a tungel to see if it is red or not, if it is smooth or not, and if it is firm or not. He knows that if a tungel is red then it is smooth.

Regulations have just been introduced which put fields of tungels into classes A, B, C, D, E; a field may be in more than one class.

Class A: All fields containing no red tungel.
Class B: All fields containing no smooth tungel.
Class C: All fields in which every red tungel is firm.
Class D: All fields in which every smooth tungel is firm.
Class E: All fields containing a red tungel which is not firm.

To test Farmer Midear’s understanding of the regulations he was asked to say which of the following statements are true.

1. If a field is in A then it is in B.
2. If a field is in B then it is in A.
3. If a field is in C then it is in B.
4. If a field is in B then it is in C.
5. If a field is in C then it is in D.
6. If a field is in D then it is in C.
7. If a field is in D then it is not in E.
8. If a field is not in E then it is in C.
9. In every field there is a tungel such that if it is not red then the field is in A.

What should Farmer Midear’s answer be?