# Enigmatic Code

Programming Enigma Puzzles

## Enigma 534: Under and over

From New Scientist #1686, 14th October 1989 [link]

The important C.A.R. Rally has just been held at Bun’s Hutch, where the track is as in the map.

There were 12 cars in the rally, numbered 1 to 12. For the start at 10:00, the cars took up the 12 lettered positions on the map, one car to each position. At 10:01 the cars all reached their first bridge, going under or over; at 10:02 the cars all reached their second bridge.

I had taken a photo at 10:01 and also at 10:02. Each photo showed all the numbers of the cars and so each bridge showed a fraction with one number over another. On one photo the fractions worked out to be 1/2, 2/3, 2/3, 2 1/3, 4, 5 1/2, and on the other 2, 2, 2, 3, 5 1/2, 7, but I forget which photo was which.

Which cars started at each of the 12 letters?

[enigma534]

## Puzzle #41: Hen party dorm

From New Scientist #3264, 11th January 2020 [link]

Ten friends have rented a dormitory for the night of a hen party. Each person picks a bed for the night before heading out on the town. At 2 am they start heading home a little the worse for wear.

Amy, the first to arrive back at the dorm, can’t remember which bed she chose, so she picks one at random. The next person to return, Bethan, heads for her own bed, but if she finds it has already been taken, she randomly picks another.

The remaining friends adopt the same approach of going to their bed if it is available and randomly picking another if it isn’t. Janice is the last to get home. What is the chance that her own bed is still empty? And was Janice more or less likely to find the bed she first chose empty than Iona, who got back just before her?

[puzzle#41]

## Enigma 976: This happy breed

From New Scientist #2131, 25th April 1998

At the end of 1991 the Society for the Protection of Our Obscure Furry Friends (SPOOFF) released a trial number of breeding pairs (born in March of that year) of the spotted tree-rat Sciurus maculatus incastus in the small forested island of Yorkiddin.

The result is a doubtful success. In 1997 they found that the island was being overrun: the local foresters were seeking compensation for damage and local rare species of birds were near extinction.

Tree-rats are born in March, and are driven from the patch of forest of their birth at the end of that year. They find a new patch, always of 2500 square metres, and immediately start breeding. They die in the fourth December of their lives. The society realised too late that every year each pair invariably produces 2 pairs of young, each of which incestuously produces 2 more pairs next year. In 1998 the number of pairs will reach nearly 6000.

The process will continue until the population reaches its limit at the beginning of 2000, when the whole forested area will have been partitioned into occupied territories.

What is the forested area in hectares? (One hectare is equal to 10,000 square metres).

[enigma976]

## Puzzle 6: On the silly side of Silly Street

From New Scientist #1057, 23rd June 1977 [link]

On the Island of Imperfection there are three tribes, the Pukkas who always tell the truth, the Wotta-Woppas who never tell the truth, and the Shill-Shallas who make statements which are alternately true and false (or false and true).

The four inhabitants of the Island with whom this story deals live in separate houses all on one side of Silly Street where the numbers of the houses are all odd, from 1 to 41 inclusive.

They speak as follows:

A.1: D’s number is one third of C’s;
A.2: D is a Wotta-Woppa.

B.1: The numbers of all our four houses are multiples of 5;
B.2: C’s number is less than A’s.

C.1: A belongs to the same tribe as I do;
C.2: A Pukka lives in No 35.

D.1: C’s number is greater than B’s;
D.2: B belongs to a more truthful tribe than I do.

You are told that there is at least one representative from each tribe, but only one Shilli-Shalla.

Find their tribes and the numbers of their houses.

[puzzle6]

## 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]

## Puzzle #40: Light bulb moment

From New Scientist #3263, 4th January 2020 [link] [link]

A tall office building is being rewired. There is a staircase, but the lift is out of action.

There are four identical-looking wires, A, B, C and D, feeding into a pipe in the ceiling of the basement. You are reasonably confident that it is those same four wires that emerge from a pipe on the top floor. Unfortunately the wires have become tangled, so it isn’t known which wire becomes 1, 2, 3 or 4.

To find out, you can join two wires together in the basement (for example A and C) and you can attach two wires at the other end to a light bulb and battery (for example 1 and 3). If the bulb lights, you have made a circuit.

Starting in the basement, what is the smallest number of light bulb flashes that you need in order to figure out which wire is which? And how many times do you need to climb the stairs?

[puzzle#40]

## Enigma 977: Walk and drive

From New Scientist #2132, 2nd May 1998

Anne, Barbara and Christine walk at 4 mph and drive at 48 mph. They have a journey of 24 miles to do but their car only takes 2 people. So Anne sets off walking, while Barbara drives so far with Christine, who then gets out and walks the rest of the journey. Barbara drives back until she meets Anne, picks her up and the drive the rest of the journey. They choose Christine’s dropping place so as to minimise the time taken by the last person to arrive.

1. How many minutes did the last person take for the journey?

Donald, Eric and Frank have to make a journey of 265 miles. They each walk at a speed which is a whole number of mph and the speed of their 2-seater car is a whole number times their walking speed and is less than 50 mph. They use the same plan as the ladies. The last person to arrive takes an odd number of hours for the journey.

2. How many hours did the last person take for the journey?

[enigma977]

## Tantalizer 414: Three men in a catastrophe

From New Scientist #964, 28th August 1975 [link]

George, Harris and I once decided to acquire a cat apiece. Dear little furry things they were, little bundles of innocence. Or so we thought before we got them home.

But original sin will out and we soon found that none of them was in the least well behaved. George’s was better behaved than the Siamese. The Persian was better behaved than Bubbles. Harris’s was better behaved than Fluff. The Tabby was not worse behaved than the Persian.

All in all we weren’t sorry when Montmorency, the dog, took a hand [paw?] and sent them caterwauling. He took a particularly satisfying bite out of Pollyanna. Whose cat was she?

[tantalizer414]

## 2019 in review

#### Happy New Year from Enigmatic Code!

There are now 1343 Enigma puzzles on the site, along with 86 from the Tantalizer series and 85 from the Puzzle series (and a few other puzzles that have caught my eye). There is a complete archive of Enigma puzzles published between January 1979 to September 1989, and from May 1998 up to the final Enigma puzzle in December 2013, which make up just over 75% of all the Enigma puzzles published. Of the remaining 450 puzzles I have 75 left to source (numbers 901 – 976).

In 2019, 103 Enigma puzzles were added to the site (and 25 Tantalizers, 25 Puzzles, and 40 others, so 193 puzzles in total).

Here is my selection of the puzzles that I found most interesting to solve over the year:

::

::

::

#### Sunday Times Teasers

I have also been collecting some old Teaser puzzles originally published in The Sunday Times on the S2T2 site, as well as accumulating my notes for more recent Teaser puzzles that I solved at the time. There are currently 232 puzzles available on the S2T2 site.

Here are some that I found interesting to solve (or revisit):

::

Between both sites I have posted 426 puzzles in total this year. I don’t expect to maintain this rate in the future.

Thanks to everyone who has contributed to the sites in 2019, either by adding their own solutions (programmatic or analytical), insights or questions, or by helping me source puzzles from back-issues of New Scientist.

## Enigma 532: Friday the 13th

From New Scientist #1684, 30th September 1989 [link]

John Thomas, who is 50 this year [i.e. in 1989], was not really superstitious till Betty left him. In fact, they were engaged on a Friday the 13th. They were married after 13 weeks of courtship on a Friday the 13th. The married couple were abroad for their Christmas vacation. Towards the end of that vacation they quarrelled and disagreed on almost everything. They were eventually separated on a Friday the 13th, after exactly 13 weeks of married life. These events made John very, very superstitious.

When did Betty leave John?

[enigma532]

## Enigma 978: The ABC brick company

From New Scientist #2133, 9th May 1998

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]

## Puzzle 7: The Woogle on the wardrobe

From New Scientist #1058, 30th June 1977

I thought Professor Knowall looked at me rather strangely as I came into the office one day.

“You don’t look, my dear Sergeant Bungle”, he said, “as though you have had a very good night’s sleep”.

I was amazed by his perception. In a flash he had seen not only that I was not feeling quite myself, but also why. The least I could do, I felt, was to give him more information about the sleepless night thats I had been having.

The last few nights had been rather windy. I had heard, as I lay in bed, an intermittent high-pitched squeak, and a regular dull thud.

I got out of bed, clasped with one hand the woogle that hangs outside my wardrobe and with the other the chumph that is loose on the top of my chest of drawers, and steadied with my foot the pollux, which is normally free to move around the floor on casters. The thud stopped, but the squeak continued.

I kept hold of the chumph, seized with my other hand the Venetian blind, and transferred my foot to the rocking chair. The squeak stopped and there was still no thud.

I then kept hold of the Venetian blind, seized the woogle once more and took my foot off the rocking chair. The thud started up again, but there was no squeak.

But what could I do? I know that the Professor would approve of my making these experiments, but not many can control woogles, chumphs, polluxes’, Venetian blinds and rocking chairs, all at once, and I had been unable to come to any conclusions as to what caused the noises.

The professor, with the information that I had given him, solved the problem in less time than it takes to steady a pollux with a foot.

What can you say about the causes of the two noises?

Happy Christmas from Enigmatic Code!

[puzzle7]

## Enigma 531: Petits fours

From New Scientist #1683, 23rd September 1989 [link]

“Four-armed is four-warmed,” declared Professor Törqui as he placed the petits fours in the oven in his lab at the Department of Immaterial Science and Unclear Physics. “There are 4444 of them: a string of 4s. By which I mean, naturally enough, a number in base 10 all of whose digits are 4. Do you like my plus fours? [*] Speaking of 10s and plus fours, you can hardly be unaware of the fact that all positive integral powers of 10 (except 10¹, poor thing) are expressible as sums of strings of 4s.”

“The most economical way of expressing 10² as a sum of strings of 4s (that is, the one using fewest strings and hence fewest 4s) uses seven 4s:”

10² = 44 + 44 + 4 + 4 + 4.

“The most economical means of expressing 10³ as a sum of strings of 4s requires sixteen 4s:”

10³ = 444 + 444 + 44 + 44 + 4 + 4 + 4 + 4 + 4 + 4.

“Now, it’s four o’clock, and just time for this puzzle: Give me somewhere to put my cakestand and I will make a number of petits fours which is an integral positive power of 10 such that the number of 4s required to write it as a sum of strings of 4s in the most economical way is itself a string of 4s.”

What is the smallest number of petits fours Törqui’s boast would commit him to baking? (Express your answer as a power of 10.)

[*] £44.44 from Whatsit Forum.

[enigma531]

## Puzzle #35, #36, #37, #38, #39: A bunch of brain teasers

From New Scientist #3261, 21st December 2019 [link] [link]

#### Puzzle #35: Christmas gifts

Q1: By the twelfth day of Christmas, my true love has given me 12 partridges in a pear tree. But which gifts have I received the most of?

Q2: I want to give all the gifts back. Starting on 26 December 2019, I am going to give one of them to my true love every day. On which date will I give them my final gift?

::

#### Puzzle #36: All squares (1)

I met Natalie the other day. She wasn’t prepared to tell me her age, but she did tell me that in the year N², she will turn N years old.

In what year was she born?

::

#### Puzzle #37: All squares (2)

Can you work out (68² – 32²)/(59² – 41²) without using a calculator?

And can you do it without having to square any of the numbers?

::

#### Puzzle #38: Meaningful matches (1)

The figure below has four equilateral triangles. Move two matchsticks to get only three equilateral triangles.

::

#### Puzzle #39: Meaningful matches (2)

The figure below is composed of 29 matchsticks. Move two matchsticks to get a correct multiplication result.

::

[puzzle#35] [puzzle#36] [puzzle#37] [puzzle#38] [puzzle#39]

## Enigma 979: Triangular and Fibonacci numbers

From New Scientist #2134, 16th May 1998 [link]

Triangular numbers are those that fit the formula n(n + 1)/2, so that the sequence starts: 1, 3, 6, 10, 15, 21, 28, …

From the first 30 triangular numbers select a set that uses each of the ten digits 0 to 9 once.

1. What are the largest and smallest numbers in your set?

In the Fibonacci sequences the first two terms are 1 and 1 and each succeeding term is the sum of the previous two terms, so that sequence starts: 1, 1, 2, 3, 5, 8, 13, …

From the first 30 Fibonacci numbers select a set that uses each of the ten digits 0 to 9 once.

2. What are the largest and smallest numbers in your set?

[enigma979]

## Tantalizer 415: Dizzy spell

From New Scientist #965, 4th September 1975 [link]

Here is a simple prescription for vertigo. Draw a 4×4 grid (16 cells in a 4×4 array). Write a V in the bottom left corner, an E in each of the two adjacent cells, and R in each of the three empty cells adjacent to an E, a T in each of the four empty cells adjacent to an R, an I in each of the three empty calls adjacent to a T, a G in each of the two empty cells adjacent to an I and an O in the top right corner.

To induce mild vertigo, work out how many ways there are of spelling VERTIGO, starting the the bottom left corner and ending at the top right, as if you were an ant taking exercise. The answer is 20. Feeling better?

Tougher exercise can induce schizophrenia. So start again with a 7×7 grid and repeat the prescription with SCHIZOPHRENIA.

How many ant-like ways are there of spelling that?

[tantalizer415]

## Enigma 530: Sudden death

From New Scientist #1682, 16th September 1989 [link]

There were eight players in the Greenchester Knock-Out Golf Championship. Unfortunately, due to rain, the whole competition had to be played on Saturday afternoon, and so it was decided to play the four first-round matches, the two semifinals and the final as sudden-death matches.

Thus in each match the two players played one hole and, if the scores were different, then the lower was the winner of that match. If the scores were equal then the played another hole with the same procedure applying, and so on, until the winner of that match was found.

After the competition the organiser listed each player’s scores for the holes (s)he had played, in order. Unfortunately he did not indicate the number of holes played in each round, but ran the scores together in a single list, as follows:

Anne: 3,3
Bern: 4,2,3,4
Chris: 4,2,3,3,3,3,3,3,2
Donald: 3,2,3,3,3
Eric: 4
Frances: 4,2,3,3,4
Grace: 3,2,3,4
Harriet: 4,2,3,3,2,4

Who beat whom in the semifinals?

[enigma530]

## Puzzle #34: Ant on a tetrahedron

From New Scientist #3259, 14th December 2019 [link] [link]

Three short-sighted spiders are clustered at the vertex of a wire frame in the shape of a tetrahedron. The spiders know that there is an ant walking around the frame, but they have no idea where it is. They will only be able to spot it when they are practically on top of it. The ant, on the other hand, has excellent eyesight and can plan its route accordingly to avoid the spiders. Given that the ant walks slightly slower than the spiders, is there a way for the ant to escape the spiders indefinitely? Or can the spiders find a strategy to be certain of catching the ant?

[puzzle#34]

## Enigma 980: Near and fair

From New Scientist #2135, 23rd May 1998 [link]

Mary stood at the side of a large pile of turnips, which she was to distribute evenly between the needy people of the area, who were standing in front of her. The rule was that if the turnips did not divide evenly between the people that Mary should go to the nearest sensible division. (If necessary, extra turnips could be added or spare turnips disposed of). Quickly, she divided the number of turnips by the number of people and found the answer was between 99 and 100, and nearer to 99. As she knew that 100 turnips would be better for people’s health she decided to carry out the division in a special way.

She announced: “Suppose everyone get 99 turnips. I have the divided the number of turnips by 99”, and she gave the answer to lots of decimal places. “Now, suppose everyone gets 100 turnips. I have divided the number of turnips by 100”, and she again gave the answer.

“If we look at our two answers, then we find that the one that is nearer to the actual number of people is when everyone gets 100 turnips. So, by the rule, that is what everyone will get”.

The distribution did not take long as there were fewer than twenty thousand turnips in the pile.

How many people were there to receive turnips, and how many turnips were there in the pile?

[enigma980]

## Puzzle 8: Division (letters for digits)

From New Scientist #1059, 7th July 1977 [link]

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

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

[puzzle8]