Enigmatic Code

Programming Enigma Puzzles

Tag Archives: by: Keith Austin

Enigma 449: He who laps last

From New Scientist #1600, 18th February 1988 [link]

“This is Hurray Talker reporting from Silverhatch on the 40-lap Petit Prix. The leaders, Hansell and Bisquet, are each going round the circuit in an incredible 59 seconds, except when they have made a pitstop. They were neck and neck for the first seven laps but then things started to go wrong. However, each pit stop has been kept down to an amazing 23 seconds. Here they come now to cross the line and complete another lap. Hansel – now. Biscuit – now. That’s just a 2-second interval. There they go off on the next lap.”

How many laps have Hansell and Bisquet completed?

[enigma449]

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Enigma 1062: Christmas present

From New Scientist #2218, 25th December 1999

Joseph the carpenter used to cut out rectangular blocks of wood which his young son Jesus would paint. The blocks always had whole number dimensions. They used to say a block was fair if the numerical values of its volume and its surface area were the same, for example the 4×5×20 block was fair as it had volume and surface area both equal to 400. They felt that with a fair block the both did the same amount of work.

As Jesus’s birthday was coming up, Joseph asked him to choose a number and he would try and cut a fair block with volume equal to that number. Jesus chose 2000 which so surprised Joseph that he asked Jesus if he thought people would remember him on his 2000th birthday. Jesus thought for a while then replied that it was hard to say, as it depended on so many things.

Can Joseph cut a fair block with volume 2000? If he can, give its dimensions. If he cannot, give the dimensions of the fair block with volume nearest to 2000.

[enigma1062]

Enigma 445: Court order

From New Scientist #1596, 21st January 1988 [link]

The draw for the Humbledon Ladies Tennis Tournament was as follows:

Enigma 445

After the tournament the umpire noticed that six ladies each lost to the lady one place below them in the above list, three ladies each lost to the lady one place above them, one lady to the lady two places below, one to the lady two places above, one to the lady three places below, one to the lady three places above, one to the lady five places below, and one to the lady six places above.

Who won the tournament and whom did she defeat in the final?

[enigma445]

Enigma 1066: Members of the clubs

From New Scientist #2222, 22nd January 2000

There are only 10 people on Small Island. However, there are many clubs, each consisting of the people with a particular interest. The island’s government will give a grant to any club with more than half the population as members. There are 12 such clubs.

The government wants to set up a committee of two so that every one of the 12 clubs has at least one member on the committee. This afternoon, the government is to look at the 12 membership lists and try to find 2 people to form the committee.

(1) This morning, before it sees the membership lists, can the government be certain that it will be able to find 2 people this afternoon?

There are 1000 people on Larger Island. The situation is similar to Small Island, except that there are 50 clubs that each have more than half the population as members. Also, the government wants to set up a committee of five so that every one of the 50 clubs has at least one member on the committee.

(2) Before the government sees the 50 membership lists, can it be certain that it will be able to find 5 people to form the committee?

The situation on Largest Island is similar to that on Larger Island, except that there are 1 million people.

(3) Before the government of Largest Island sees its 50 membership lists, can it be certain that it will be able to find five people to form its committee?

[enigma1066]

Enigma 442a: Hark the herald angels sing

From New Scientist #1592, 24th December 1987 [link]

“Have a mince pie.”

“Thanks. How did the carol singing go this evening?”

“Very well indeed. There were 353 of us. We started from the village church at 6:00pm and arrived here at the hall some time ago. Have a look at this map here on the wall.

Enigma 442a

“We divided into groups and between us we covered every one of the 12 roads once. For each road, the group would enter at one end, sing the carols as they walked along the road, and leave at the other end. At each of the five junctions, all the groups due to arrive at that junction would come together and then re-divide before setting off again.”

“Did you sing many carols?”

“On each road, every member of the group would sing one verse as a solo. A verse takes one minute to sing.”

“Did people get cold waiting at the junctions for the other groups to join them?”

“No. That is the marvellous thing. We had divided into groups so that, at each junction, all the groups for that junction arrived at precisely the same time. Similarly, all the groups arrived at the hall at the same time.”

“You were very fortunate — a little miracle.”

“Don’t forget,” said the vicar who had been standing with us, “it is Christmas Eve.”

What time did the singers arrive at the hall and what was the total number of verses that they sang?

[enigma442a] [enigma442]

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.

[enigma441]

Enigma 1070: Time to work

From New Scientist #2226, 19th February 2000 [link]

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?

[enigma1070]

Enigma 1072: Into three piles

From New Scientist #2228, 4th March 2000 [link]

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?

[enigma1072]

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.

[enigma437]

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?

[enigma432]

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!

[enigma1082]

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?

[enigma428]

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.

[enigma1084]

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:

9/10,
25/27,
14/15,
17/18,
18/19,
19/20,
29/30,
31/32,
1,
68/67,
36/35,
28/27,
12/11,
7/6,
9/7,
7/5,
3/2,
5/3.

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?

[enigma424]

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.

[enigma1089]

Enigma 419: Painting by numbers

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

Instructions

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.

Example

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?

[enigma419]

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?

[enigma415]

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?

[enigma1097]

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?

[enigma411]

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}
[3]

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.

[enigma1101]