# Enigmatic Code

Programming Enigma Puzzles

## Enigma 541: A roundabout route

The following puzzle was sent in by a young schoolboy:

“Five businessmen (Adam, Brian, Clive, David and Edward) all live in Glasgow. On their way to work they all approach the same roundabout from the same direction and each takes a different one of the five possible other exits from the roundabout (labelled A, B, C, D, E in clockwise order from their entry).

The exits lead to Aberdeen, Berwick, Carlisle, Dundee and Edinburgh. In each case the letter of the exit, the initial of the man using that exit, and the initial letter of the destination are all different.

David leaves at an exit labelled with a vowel, Adam leaves the roundabout at the exit before Edward, and the man going to Edinburgh drives past Clive’s exit and leaves three exits later.

What is the label on the Aberdeen exit? Who takes the Dundee exit?”

Unfortunately, this brave attempt does not have a unique answer so I told the young man to resubmit it with an extra clue of the type “Exit ___ does not lead to ___”. He did this and it did then make an acceptable Enigma.

[enigma541]

## Enigma 969: Same from all angles

From New Scientist #2124, 7th March 1998

For the purposes of this Enigma I shall define a “palindromic angle” as one whose number of degrees is a whole number less than 180 and is such that the number remains the same if its digits (or digit) are written in reverse order.

I have drawn an irregular hexagon and a line across it in a similar fashion to the one shown here, however the given picture is not to scale. In my figure, the eight marked angles are different palindromic angles, and in fact more than two of them are acute.

What (in increasing order) are the six angles of my hexagon?

[enigma969]

## Enigma 972: Fifty-fifty

From New Scientist #2127, 28th March 1998

Bunko, Jack and Patience have a pack of cards that consists of fewer than one hundred cards which are numbered consecutively: 1, 2, 3, 4, 5, 6, …

On one occasion recently they were each given one of the cards (without the other two players seeing which).

Bunko said: “There’s an exactly 50:50 chance that my card is the highest of our three”.

Then Jack added: “In that case there’s an exactly 50:50 chance that, in decreasing order, the cards are Bunko’s, Patience’s then mine”.

How many cards are there in the pack? And what was Jack’s card?

[enigma972]

## Enigma 537: The three sisters

Sarah, Tora and Ursula are three sisters. One of them is honest, one always lies, and the third is simply unreliable.

Tora told me that the youngest of the three is fatter than the liar, and the oldest of the three told me that the fattest is older than the honest one. But Ursula and the thinnest sister both agreed that the oldest sister always lies.

The fattest sister and the oldest sister both agreed that Sarah always lies. Then the youngest sister whispered to Sarah who then claimed that the youngest had said that Ursula was the fattest.

Who is the honest sister?
Who is the fattest sister?
Who is the youngest sister?

[enigma537]

## Enigma 533: An odd enigma

From New Scientist #1685, 7th October 1989 [link]

In this long division sum, in the dividend and divisor, I’ve replaced digits consistently with letters, with different letters for different digits, and left gaps in all other places where digits should be:

You don’t actually need any more clues, but I can tell you that this ENIGMA is odd.

What is this odd ENIGMA?

[enigma533]

## Enigma 978: The ABC brick company

From New Scientist #2133, 9th May 1998 [link]

The ABC brick company prides itself on making unique toys. It has just produced a range of wooden bricks, all of the same size, in the shape of a tetrahedron (a solid with four equilateral-triangle faces). Each of the four faces on every tetrahedron is painted in one of the company’s standard colour range. For example, one of the bricks has one yellow face, two blue faces, and a green face. The company ensures that each tetrahedron is different — there is no way of rotating one to make it look like another. With that restriction in mind, the company has manufactured the largest possible number of these bricks.

To add to the uniqueness of the toys, each brick is placed in an individual cardboard box with the letters “ABC” stencilled on it. Then using the same standard range of the company’s colours, an artist paints each of the letters on the boxes. For example, one has a red “A”, a blue “B”, and a red “C”. No two of the colourings of the ABCs are the same, and, with that restriction in mind, once again the company has produced the largest possible number of boxes.

By coincidence, there are just enough boxes to put one of the tetrahedra in each.

How many colours are there in the company’s standard range?

#### News

There are now 450 Enigma puzzles remaining to post, which means that 75% of all Enigma puzzles are now available on the site.

[enigma978]

## Enigma 528: An enigma for U

From New Scientist #1680, 2nd September 1989 [link]

Here are just three rows from our local football league table at the end of the season, after each team in the league has played each of the others once. The teams have been put in alphabetical order here.

There are three points for a win and one for a draw and, in the table, digits have been consistently replaced by letters with different letters used for different digits.

Please find AGAIN. And tell me who (if anybody) won when City played Albion.

#### News

It’s the 8th anniversary of Enigmatic Code, and this puzzle brings the total number of Enigma puzzles posted to 1335, which is 75% of the 1780 Enigma puzzles published in New Scientist. There is a complete archive of puzzles from the first Enigma puzzle in February 1979 up to September 1989, and also from June 1998 up to the final Enigma puzzle in December 2013. There are 457 Enigma puzzles remaining to post.

Also available are puzzles from the Puzzle series, which were published in New Scientist before Enigma started. There is a complete archive available from July 1977 until the end of the Puzzle series in February 1979 (83 puzzles). There are 7 puzzles in this series remaining to post.

And before that was the Tantalizer series of puzzles, of which there is a complete archive from September 1975 up to the end of the Tantalizer series in May 1977 (84 puzzles).

Earlier in 2019 New Scientist started publishing a new series of puzzles (the “Puzzle #” series), and I have been posting these to the site, along with my notes, as they became available.

I have also been posting my notes on Sunday Times Teaser puzzles at the S2T2 site, and there are currently 227 puzzles available there.

So between the two sites there are currently 1766 puzzles available, which is almost the total number of Enigma puzzles published.

[enigma528]

## Enigma 982: Break even

From New Scientist #2137, 6th June 1998 [link]

In a recent frame of snooker “Earthquake” Endry and “Polly” Parrot had a close contest.

Earthquake went first, and each time either player visited the table the number of balls potted was the same (and the same for both players). And each time Earthquake potted a red he followed it by attempting to pot one particular colour. And each time Polly potted a red he followed it by attempting to pot one particular colour.

Building up to a grand climax, Earthquake potted the pink but was still behind. He then potted the black to win.

That’s not enough information for you to be able to answer the following question, but if I told you whether or not two consecutive pots were ever the same colour then you would have enough information to answer the question.

What was the final score in points?

(Remember, in snooker there are 15 reds. The potting of a red earns one point and enables the player to try to pot one of the other six colours, earning from 2 to 7 points. A successful pot means that the player can try for another red, and so on. The reds stay down but the other colours are put back on the table. If an attempt to pot a ball fails then the other player has a turn at the table. When all 15 reds have gone and all 15 attempted pots at colours are completed, the six remaining colours are potted in order, and this time they too stay down. No other rules concern the frame described in this Enigma).

[enigma982]

## Enigma 524: Spot check

From New Scientist #1676, 5th August 1989 [link]

I’m in the middle of a game of dominoes and I’m looking at the four dominoes laid end-to-end on the table.

If I replace the number of spots by digits and then read off the four-figure number formed by the first two dominoes, I get a perfect square. (For example, had the dominoes been:

then I’d have read the number 1220).

The four-figure number formed by the middle two dominoes (which would be 2005 in the above example) is a prime.

And the total number of spots on the four dominoes is a perfect square.

What are the four dominoes.

[enigma524]

## Enigma 986: Not the World Cup

From New Scientist #2141, 4th July 1998 [link]

Our local football league organised a mock World Cup championship. The teams (less than a thousand altogether!) were divided into a number of equal sized groups. Within each group, each team played each of the other teams. Then just the best two from each group qualified for the final stages in which all the qualifiers played a knock-out competition.

The knock-out competition started with all those qualifying teams being paired off for games, with the winners of those games going through to the next round. Those winners were then paired off for games, with the winners of those games going through to the next round, and so on, eventually leading to four teams playing in the semi-finals and two in the final. In addition, the two semi-finals losers played for third and fourth place.

It turned out that the number of games played by the eventual champions was twice the average number of games played overall by all the teams.

How many teams entered our World Cup championship?

[enigma986]

## Enigma 519: Fibonacci thimbles

From New Scientist #1671, 1st July 1989 [link]

One year, on my birthday, I started a collection of thimbles. The following birthday I added to my collection, which went from strength to strength. In all subsequent years when I counted the thimbles on my birthday the total had increased from the previous year’s total by a number equal to the total I had on my birthday the year before that. (So, for example, my 1983 total equalled my 1982 total added to my 1981 total).

Now, by coincidence, my daughter was born on my birthday. And, with my collection growing following the described pattern, on our birthday in 1983 the number of thimbles I owned had reached exactly four times my daughter’s age on that day. On my birthday this year the total of thimbles was four times my age. On only one other occasion has the total been divisible by four, and that was in the year my son was born.

How many thimbles were there in my collection on my birthday this year? How many (if any) did I have the day my daughter was born?

[enigma519]

## Enigma 991: Add it up

From New Scientist #2146, 8th August 1998 [link]

Throughout the following equations each letter stands consistently for either a non-zero digit or for one of the arithmetic symbols ×, ÷, + and –, with different letters being used for different symbols/digits.

When you have substituted the digits and symbols both equations make sense working from left to right (so that, for example, 5 – 2 × 23 = ((5 – 2) × 23) = 69), are correct, and, as our title implies, a + features somewhere.

AN = ENIGMA

Find the value of AND.

[enigma991]

## Enigma 515: Foreign ties

From New Scientist #1667, 3rd June 1989 [link]

The Anglo-Slovak club had its meeting last week. Those present were Tom, Vyctur, Ted, Tago, Ray, Min, Wex, Olav, Russ and Cy.

Some of the members stood up and took part in an old Slovakian dance, rather like a Morris dance. The dancers stood around the floor with no three in a straight line and between each pair a taut piece of ribbon was stretched across the floor. Some ribbons were pink and the rest were blue. I noticed that there was a pink ribbon between two of them precisely when their Christian names had an odd number of letters in common. (So, for example, had a Jane, David and Victor been dancing, there would have been a pink ribbon from Jane to David, a blue from David to Victor, and a blue from Jane to Victor).

As soon as I saw how many dancers there were I realised that two of the ribbons would have to cross. But they had arranged themselves in such a way that there was no pink triangle and no blue triangle of ribbons.

Who was dancing?

[enigma515]

From New Scientist #2150, 5th September 1998 [link]

This is the layout of the digit buttons on my telephone. My boss’s telephone number uses each of the ten digits once and it starts with 0. Furthermore each pair of adjacent digits in her telephone number is also adjacent (horizontally, vertically or diagonally) on the telephone keypad. (By coincidence my own number, 0895632147, has the same properties).

I have just looked through my boss’s telephone number and written down a list of all the two-figure numbers that can be seen in it by reading a pair of adjacent digits (which would be 89, 95, 56… in my number). In that list, some of the numbers are special in that they consist of two digits which are also consecutive (such as 89, 56, 32 and 21 in my number). I have worked out the product of those special numbers and it is the year in which my boss will be 50. And if I look at the number of the year in which she was born, no two of its adjacent digits are adjacent (either way round) in her telephone number.

What is her telephone number?

Enigma 486 is also called “Number please”.

[enigma995]

## Enigma 511: Double, double …

From New Scientist #1663, 6th May 1989 [link]

I wrote an odd number on the board and asked the class how many numbers (including the original number itself) could be made by writing exactly the same digits but in different orders. (For example, if the number had been 5051, the answer would have been nine, namely 5051, 5015, 5105, 5150, 5501, 5510, 1055, 1505 and 1550).

Clever Dick got the right answer immediately, so to keep him busy I told him to repeat the exercise with exactly double my original number.

“That just doubles the number of ways, Miss,” he reported.

I told him to double again and repeat the exercise, and again he reported “That doubles the number of ways yet again, Miss.”

So I told him to double the number yet again and to repeat the exercise with the four-figure answer.

“It’s doubled the number of ways again, Miss,” he replied and, as always, he was quite right.

What number did I write on the board?

[enigma511]

## Enigma 999: Combined celebrations

From New Scientist #2154, 3rd October 1998 [link]

To celebrate next week’s 1000th edition of Enigma, we each made up an Enigma. Each one consisted of four clues leading to its own unique positive whole number answer. In each case none of the four clues was redundant. To avoid duplication, Keith made up his Enigma first and showed it to Susan before she made up hers.

The two Enigmas were meant to be printed side-by-side but the publishers have made a (rare) error and printed the clues in a string:

(A) It is a three-figure number;
(B) It is less than a thousand;
(C) It is a perfect square;
(D) It is a perfect cube;
(E) It has no repeated digits;
(F) The sum of its digits is a perfect square;
(G) The sum of its digits is a perfect cube;
(H) The sum of all the digits which are odd in Keith’s answer is the same as the sum of all the digits which are odd in Susan’s.

Which four clues should have formed Keith’s Enigma, and what was the answer to Susan’s?

#### News

There are now 1300 Enigma puzzles available on the site (or at least 1300 posts in the enigma category). There are 492 Enigma puzzles remaining to post.

There are currently also 76 puzzles from the Tantalizer series, 75 from the Puzzle series and 13 from the new Puzzle # series of puzzles that have been published in New Scientist which together cover puzzles from 1975 to 2019 (albeit with some gaps).

I also notice that the enigma.py library is now 10 years old (according to the header in the file – the creation date given coincides with me buying a book on Python). In those 10 years it has grown considerably, in both functionality and size. I’m considering doing a few articles focussed on specific functionality that is available in the library.

[enigma999]

## Enigma 1001: What the hex?

From New Scientist #2156, 17th October 1998 [link]

In this hexagon of circles I’ve written some digits:

Reading the six sides, clockwise, as three-figure numbers you get 187, 714, 425, 527, 799, and 901, all of which are multiples of 17. Your task today is to write a new collection of non-zero digits in the circles, with no two adjacent digits the same, so that the six three-figure numbers are all different multiples of some particular two-figure number, the number in the top row being twice that two-figure number.

What are the numbers in your hexagon?

[enigma1001]

## Enigma 1003: Dicey names

From New Scientist #2158, 31st October 1998 [link]

I have three cube-shaped dice with a letter on each of the faces. I throw the three [dice] and then try to arrange the letters showing uppermost into one of my friend’s names. I then turn all three dice right over and try to make one of my other friend’s names by arranging the new uppermost letters (which were previously on the bottom).

In this way I can get various pairs of names. For example I can get TOM paired with ANN, or I can get PIP with JON, or PAT with BOB, or ABE with DOT.

I asked my nephew to try all possible arrangements of the dice and see which of my friends’ names could be paired with BON. He found that BON could be paired off with at least two of the names BAP, BIP, MAT, NAT, PEN, PIT, TED and TIM.

Which of those can it be paired off with, and which other five letters are on the same die as the J?

[enigma1003]

## Enigma 506: Remember an Enigma

From New Scientist #1658, 1st April 1989 [link]

Alan, Brian, Charles and David were having a celebration: one or more of them was celebrating a birthday that day. My brother called in to see them and had a chat with each of them. They each made a couple of statements. Each boy was either honest or always told lies. Even knowing this my brother was able to work out which boy (or boys) was celebrating a birthday.

When my brother told me this story, I thought it would make a good “Enigma”, so I asked him for the details of the statements which the boys had made. Unfortunately, he couldn’t remember exactly what they had said.

Either Alan said “It’s my birthday”, or he said “It’s not my birthday”; and then he added, “but Charles is a liar”.

Brian named the one person whose birthday it was, and added that exactly half the boys were liars.

My brother forgets the name which Brian quoted but remembers that Charles claimed that that particular boy was also honest, and added that both Alan and Brian were celebrating their birthdays.

David said that there were two or more boys celebrating their birthdays and either he said that boys always lie on their birthday or he said that boys whose birthday it is today never lie on their birthdays.

Whose birthday was it?

What did David say that boys do on their birthdays?

[enigma506]

## Enigma 1006: Booklist

From New Scientist #2161, 21st November 1998 [link]

I have just been looking at the top six best-selling books listed in today’s paper. They are numbers from 1 (for the best selling book) to 6, and after each book its position in last week’s list is given.

It turns out that the same six books were in the list last week. For each of the books I multiplied the number of last week’s, and I got six different answers.

Now, if you asked my the following questions — then you would get the same answer in each case:

1. How many primes are there in that list of six products?
2. How many perfect squares are there in that list of six products?
3. How many perfect cubes are there in that list of six products?
4. How many odd numbers are there in that list of six products?
5. How many of the six books are in a higher place this week than last?

For the books 1-6 this week, what (in that order) were their positions last week?

[enigma1006]