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Programming Enigma Puzzles

27 May 2020

Posted by on **From New Scientist #951, 29th May 1975** [link]

Professor Pfiffelsammler has been looking into the motivation of cinemagoers. Recently he accosted a line of middle-aged men waiting patiently in the rain to get into “The Way of All Flesh”. The man at the head of the queue said: “There are 39 men behind me. The man at the rear is only here for the smut”. Each other man said: “I am here for the sake of the film’s aesthetic merits. The man in front of me is only here for the smut”.

One tenth of the queue having been admitted, the new front man declared: “There are 71 men behind me”. One ninth of those still waiting were then let in and the new front man said: “There are 15 men behind me”. One eighth of those still waiting were then admitted and the new front man asserted: “There are 27 men behind me”. There were less than 100 men in the queue at the start of the investigation and no one had joined or left (except those admitted into the cinema). Naturally, all and only those there for the aesthetics had spoken the truth. No one, of course, came both for the aesthetics and the smut.

Exactly how man men were there in the queue originally?

There are now 100 *Tantalizer* puzzles on the site. This means there is a complete archive of *New Scientist* puzzles from May 1975 – February 1990 and from December 1997 – December 2013, giving a total of about 1569 puzzles on the site. There are 414 *Enigma* puzzles remaining to fill in the gap from 1990 – 1997, and there are about 220 remaining *Tantalizer* puzzles in the *Google Books* archive.

[tantalizer401]

15 May 2020

Posted by on **From New Scientist #2115, 3rd January 1998**

Harry and Tom were trying to find a set of three 3-digit perfect squares which between them used nine different digits.

Harry found such a set and also discovered that if he took his one unused digit and the three digits of one of his squares he could arrange them to form a 4-digit perfect square.

(1) What was this 4-digit perfect square?

The best that Tom could manage was to find a 2-digit perfect square and two 3-digit perfect squares which between them used eight different digits. But he also discovered that if he took either one of his two unused digits and the digits of either of his two 3-digit squares he could arrange them to form a 4-digit perfect square.

(2) List in ascending order the four 4-digit perfect squares that Tom could form.

This puzzle completes the archive of *Enigma* puzzles from 1998. There is now a complete archive of *New Scientist* puzzles from the start of 1998 to the end of 2013, and also from June 1975 to January 1990, a total of 1565 puzzles available. There are 414 *Enigma* puzzles remaining to post.

[enigma960]

13 April 2020

Posted by on **From New Scientist #1696, 23rd December 1989** [link]

Amy, Beth, Jo and Meg decided to give each other pots of marmee-lade for Christmas. Each girl made a number of pots and then divided her pots into three piles, which were not necessarily equal; then she wrapped up each pile, labelled each parcel, and put them under the Christmas tree. The total number of pots involved was between 50 and 100.

We will let the letters A, B, C, …, I stand for the digits 1, 2, 3, …, 9 in some order. Amy gave D/F of her pots to Jo, and C/B to Meg. Beth gave H/G of her pots to Amy, H/A to Jo, and D/G to Meg. Jo gave F/G of her pots to Amy. Meg gave A/B of her pots to Beth, and D/H to Jo.

On Christmas day, each girl opened the three parcels she had received. Amy received H/E of her pots from Beth, and F/E from Jo. Jo received D/I of her pots from Amy, D/H from Beth and D/A from Meg.

Note that all the fractions were in reduced form before letters were substituted (1/2 and 2/3 are in reduced form, whereas 4/8 and 6/9 are not).

What was the total number of pots that were given?

This puzzle completes the archive of *Enigma* puzzles from 1989. There is now a complete archive of *New Scientist* puzzles from July 1975 to December 1989, and from February 1998 to December 2013, a total of 1553 puzzles. There are 423 *Enigma* puzzles remaining to post.

[enigma544b] [enigma544]

17 January 2020

Posted by on I won’t be able to add new puzzles or comments to the site for the next few days.

Hopefully normal service will be resumed shortly.

1 January 2020

Posted by on 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:

**Enigma 484: Who knows?****Enigma 493a: Little donkey****Enigma 493b: Christmas cards****Enigma 500: Child’s play****Enigma 501: A reciprocal arrangement****Enigma 508: A colourful deception****Enigma 517: Walk in the dark****Enigma 527: Sum secret**

::

**Enigma 985: 57 varieties**(see also:**Teaser 1986: Protracted calculation**)**Enigma 988: Cards high and low****Enigma 991: Add it up****Enigma 999: Combined celebrations****Enigma 1003: Dicey names****Enigma 1004: The art of cubes****Enigma 1009: Squared square****Enigma 1011: The ribbon’s reach****Enigma 1018: Half-time****Enigma 1025: A score or more**

::

**Tantalizer 434: Limited editions****Tantalizer 435: Compleat idiots****Tantalizer 437: Miniatures****Tantalizer 438: Spring collection****The Prisoner of Benda**

::

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):

**Brain-Teaser 475****Brain-Teaser 479****Brain-Teaser 507****Brain-Teaser 511****Brainteaser 1568: Back to the future****Brainteaser 1631: Ali Baba’s boxes****Brainteaser 1747: Lottery charges****Brainteaser 1757: Enigma variations****Brainteaser 1782: Unwinding****Teaser 2817: Our secret****Teaser 2821: Magic cards****Teaser 2835: Jeweller’s rouge****Teaser 2848: Coffee breaks****Teaser 2903: What’s my card?****Teaser 2913: Return tickets****Teaser 2948: A hardy annual**

::

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*.

27 December 2019

Posted by on **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?

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

[enigma978]

2 December 2019

Posted by on **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.

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]

2 August 2019

Posted by on **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?

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]

12 July 2019

Posted by on **From New Scientist #2157, 24th October 1998** [link]

Albion have been playing in Europe, where the result of any fixture is decided by the aggregate scores achieved by the teams on two matches, each team being at home for one match and away for the other match. If the aggregate scores over the two matches are equal the rule is that the team that has scored more goals away from home wins.

Albion played in and won five fixtures in Europe. In four of them the aggregate scores were equal and Albion won each time on the “away goals” rule, even though in their five away matches Albion scored fewer goals than in their five home matches.

No two matches out of the ten that Albion played had the same score and no team scored more than three goals in any of the matches.

What were the scores in Albion’s five home matches? Give each score in the form

x-y, with Albion’s score first in each match.

I was going to mention that there are now “only” 500 *Enigma* puzzles left to post, but when I counted it up it looked like there were “only” 498 left, so I should have mentioned this with **Enigma 1003**. Although it is hard to be exact (there are duplicate puzzles, corrections, and sometimes extra puzzles at Christmas) I think it is safe to say that there are about 1294 *Enigma* puzzles posted so far, and about 498 left, so we are about 72% of our way through the *Enigma* puzzles.

[enigma1002]

3 May 2019

Posted by on **From New Scientist #2168, 9th January 1999** [link]

In this sum each letter represents a different digit. The same letter represents the same digit wherever it appears and no number starts with a zero.

What is the 5-digit number represented by EIGHT?

This is the first puzzle that was published in 1999, so there is now a complete 15 year archive of *Enigma* puzzles from the start of 1999 to the final *Enigma* puzzle published in December 2013. There is also a complete 11 year archive of earlier puzzles from October 1977 to January 1989. As well as *Tantalizer* puzzles from 1976 and 1977. This brings the total number of archived puzzles to over 1400. I will continue to expand the archive by posting puzzles on a regular schedule.

[enigma1012]

1 May 2019

Posted by on **From New Scientist #981, 1st January 1976** [link]

To shake down the plum pud, the five adults held three post-prandial athletic events. Each competitor scored the number of the place gained in each event, with the aim of totalling as few points as possible overall. Thus Uncle Arthur came second in the hop and scored 2 points for it. There were no ties in any event or in the overall totals and no one took the same place in two or more events.

Aunt Barbara, although bottom in one event, was top at skipping, Mother having been forced down to third place by a fit of hiccoughs. Father did better than Uncle Charlie at hopping. Uncle Arthur did not win the jumping. Mother did better at jumping than at hopping. Aunt Barbara was not second overall. The overall winner did not win the hopping.

As your post-prandial exercise, would you care to list the order in each event?

The puzzle can be solved as presented, but has two solutions. To arrive at the published single solution we seem to need an extra fact — *“Uncle Arthur finished in third place overall”*.

This puzzle completes the archive of *Tantalizer* puzzles from 1976. There is a full archive from this puzzle to the final *Tantalizer* puzzle in May 1977 (when the *Puzzle* series started).

[tantalizer430]

1 April 2019

Posted by on **From New Scientist #1644, 24th December 1988** [link]

The five couples in Yuletide Close send cards to some of their neighbours. Some of them told me who (apart from themselves) send cards.

Alan:“The cards not involving us are the ones exchanged between Brian’s and Charles’s houses, the ones exchanged between Brian’s and Derek’s, the card from Charles to Derek (or the other way round, I’m not sure which) and the card from Brian to Eric (or the other way round).”

Brenda:“Apart from our cards, Alice and Emma exchange cards, as do Dawn and Christine, and Dawn sends Emma one.”

The Smiths:“The cards not involving us are the ones exchanged by the Thomases and the Unwins, those exchanged by the Williamses and the Vincents, the one from the Thomases to the Williamses and one between the Unwins and the Vincents (but I forget which way).”

No 3:“Nos 1 and 5 exchange cards, one card passes between Nos 2 and 4 (I don’t know which way) and No 2 sends one to No 1.”Charles Thomas receives the same number of cards as he sends. On Christmas Eve, he goes on a round tour for drinks. He delivers one of his cards, has a drink there, takes one of their cards and delivers it, has a drink there, takes one of their cards and delivers it, has a drink there, takes one of their cards and delivers it, has a drink there, collects the card from them to him and returns home, having visited every house in the close.

Name the couples at 1-5 (for example: 1, Alan and Brenda Smith; 2, …).

**Enigma 1321** is also called “Christmas cards”.

This completes the archive of *Enigma* puzzles from 1988. There is now a complete archive from the start of *Enigma* in 1979 to the end of 1988, and also from February 1999 to the final *Enigma* puzzle at the end of 2013. There are 1265 *Enigma* puzzles posted to the site, which is around 70.8% of all *Enigma* puzzles published.

[enigma493b] [enigma493]

1 January 2019

Posted by on There are now 1,239 *Enigma* puzzles on the site, along with 61 from the *Tantalizer* series and 60 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 1988, and from May 1999 up to the final *Enigma* puzzle in December 2013, which make up about 69.3% of all the *Enigma* puzzles published. Of the remaining 553 puzzles I have 125 left to source (numbers 900 – 1024).

In 2018, 105 *Enigma* puzzles were added to the site (and 26 *Tantalizers,* 26 *Puzzles*, and 3 others, so 160 puzzles in total).

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

**Enigma 432: Holiday on the islands****Enigma 442a: Hark the herald angels sing****Enigma 460: Tear me off a strip****Enigma 476: A curious question****Enigma 479: Road island**

::

**Enigma 1072: Into three piles****Enigma 1069: Time for elevenses****Enigma 1066: Members of the clubs****Enigma 1064: Low score draw****Enigma 1060: In order to solve…****Enigma 1057: Recycled change****Enigma 1041: Lucky sixes****Enigma 1037: The perfect shuffle****Enigma 1035: Connected numbers**

::

**Puzzle 52: Football on the Island of Imperfection****Tantalizer 448: Love and hate****Tantalizer 447: Marching order****Grid puzzle****Sunday Times Teaser 2503****Sunday Times Teaser 2907: Combinatorial cards**

::

Thanks to everyone who has contributed to the site in 2018, either by adding their own solutions (programmatic or analytical), insights or questions, or by helping me source puzzles from back-issues of *New Scientist*.

20 August 2018

Posted by on **From New Scientist #2204, 18th September 1999** [link]

A square field has its sides running north-south and east-west. The field is divided into an 8 × 8 array of plots. Some of the plots contain cauliflower. A line of plots running west to east is called a row and line of plots running north to south is called a column.

John selects a row and walks along it from west to east, writing down the content of each plot as he passes it; he writes E to denote an empty plot and C to denote a plot containing cauliflower; he writes down EECECCEC. He repeats this for the other seven rows and writes down ECEECCCE, ECECEECC, ECCECCEE, CEECEECC, CECECECE, CECCECEE and CCECEEEC. The order in which John visits the rows is not necessarily the order in which they occur in the field.

Similarly, Mark selects a column and walks along it from north to south, writing down the content of each plot as he passes it; he writes down EECECCCE. He repeats this for the other seven columns and writes down EECCEECC, ECECECEC, ECCECEEC, CEECCECE, CECECECE, CCEEECEC and CCECECEE. The order in which Mark visits the columns is not necessarily the order in which the occur in the field.

Draw a map of the field, showing which plots contain a cauliflower.

**Enigma 1248** was also called **“Rows and columns”**.

There are now 1200 *Enigma* puzzles on the site (although there is the odd repeated puzzle, and at least one puzzle published was impossible and a revised version was published as a later *Enigma*, but the easiest way to count the puzzles is by the number of posts in the “*enigma*” category).

There is a full archive of *Enigma* puzzles from **Enigma 1** (February 1979) to **Enigma 461** (May 1988), and of the more recent puzzles from **Enigma 1048** (September 1999) up to the final *Enigma* puzzle, **Enigma 1780** (December 2013). Which means there are around 591 *Enigma* puzzles to go.

Also on the site there are currently 53 puzzles from the *Tantalizer* series, and 50 from the *Puzzle* series, that were published in **New Scientist** before the *Enigma* series started.

Happy Puzzling!

[enigma1048]

20 July 2018

Posted by on **From New Scientist #1608, 14th April 1988** [link]

In the following division sum, letters are substituted for digits. The same letter stands for the same digit wherever it appears, and different letters stand for different digits.

Rewrite the sum with letters replaced by digits.

This puzzle brings the total number of *Enigma* puzzles on the site to 1192, which means there are now more than 2/3 of all *Enigma* puzzles published in *New Scientist* on the site. There is a full archive of puzzles from October 1999 to the final *Enigma* puzzle in December 2013 (728 puzzles), and also a full archive from the first *Enigma* puzzle in February 1979 up to this puzzle from April 1988 (462 puzzles — there were sometimes multiple puzzles at Christmas). This leaves around 600 puzzles to be posted. Thanks to the ongoing efforts of Hugh Casement I have been able to acquire the text for most of these remaining puzzles (I have 134 left to source), so I can continue to keep posting them. There are also 48 puzzles on the site from the *Puzzle* sequence (with 43 left to go), and 51 puzzles from the *Tantalizer* sequence (I think I will be able to source around 268 more of these). Happy Puzzling!

[enigma457]

30 April 2018

Posted by on **From New Scientist #2220, 8th January 2000** [link]

You play this game by first drawing 20 boxes in a continuous row. You then draw a star in each box in turn, in any order. Each time you draw a star you earn a score equal to the number of stars in the unbroken row [of stars] that includes the one you have just drawn.

Imagine that you have already drawn eleven stars as shown below, and you are deciding where to place the twelfth.

Drawing the next star in box 1 would score only 1 point, in box 11 it would score 2 points. A star in box 2, 5 or 6 would score 3 points, and in box 9, 12 or 19 it would score 4 points. Drawing the star in box 16 would score 6 points.

Your objective is to amass the lowest possible total for the 20 scores earned by drawing the 20 stars.

What is that minimum total?

This puzzle completes the archive of *Enigma* puzzles from 2000. There are now 1169 *Enigma* puzzles available on the site. There is a complete archive from the beginning of 2000 until the end of *Enigma* in December 2013 (14 years), and also from the start of *Enigma* in February 1979 up to January 1988 (10 years), making 24 years worth of puzzles in total. There are 623 *Enigma* puzzles remaining to post (from February 1988 to December 1999 – just under 11 years worth), so I’m about 62% of the way through the entire collection.

[enigma1064]

6 April 2018

Posted by on **From New Scientist #1592, 24th December 1987** [link]

After our successful pantomime production in which I played the leading lady, I gave my little costarring helpers some gifts from a big bag of different trinkets, and they each got a different number and none were left.

To make it fairer I gave each helper 10p for each gift that he

didn’tget and deducted 40p for each gift that hedidget, but that still gave each of them some 10p coins as well as some gifts. It cost me £12.60 in addition to the gifts.What was the highest number of gifts received by any helper (that little fellow got less than 50p cash)?

What part was I playing?

This puzzle completes the archive of *Enigma* puzzles from 1987. There is now a complete archive from the start of *Enigma* in February 1979 to the end of 1987, and also from February 2000 to the final *Enigma* puzzle in December 2013. Making 1162 *Enigma* puzzles posted so far, which means there are about 626 left to post.

[enigma442b] [enigma442]

31 December 2017

Posted by on There are now 1,134 *Enigma* puzzles on the site, along with 35 from the *Tantalizer* series and 34 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 1987, and from May 2000 up to the final *Enigma* puzzle in December 2013, which make up about 63.3% of all the *Enigma* puzzles published. Of the remaining 654 puzzles I have 152 left to source (numbers 891 – 1042).

In 2017, 105 *Enigma* puzzles were added to the site (and 30 *Tantalizers* and 28 *Puzzles*, so 163 puzzles in total). Here is my selection of the puzzles that I found most interesting to solve over the year:

**Enigma 381: Island airlines****Enigma 382: Dice****Enigma 383: Stop watch****Enigma 391b: Christmas recounted****Enigma 409: Hands and feet****Enigma 413: Quargerly dues****Enigma 415: Buses galore****Enigma 426: Time and again****Enigma 429: Professor Quark**

::

**Enigma 1134: Luck be a lady****Enigma 1132: Phone back****Enigma 1127: Lights out**(see also:**Enigma 1137**)**Enigma 1126: Enigmatic dice****Enigma 1124: Classy glass****Enigma 1112: Patio zones****Enigma 1110: Dots and lines****Enigma 1101: Disappearing numbers****Enigma 1097: Chessboard triangles****Enigma 1091: One’s best years****Enigma 1087: Egyptian triangles****Enigma 1085: Cut and run****Enigma 1084: 1-2-3 triangles****Enigma 1082: End-of-season blues**

::

**Tantalizer 490: Diplomatic niceties****Tantalizer 484: Blockwork****Tantalizer 474: Desert crossing****Puzzle 82: A cross number**

::

I have continued to maintain the **enigma.py** library of useful routines for puzzle solving. In particular the `SubstitutedExpression()` solver and `Primes()` class have increased functionality, and I have added the ability to execute *run files*, in cases where a complete program is not required. The `SubstitutedDivision()` solver is now derived directly from the `SubstitutedExpression()` solver, and is generally faster and more functional than the previous implementation.

I’ve also starting putting my **Python** solutions up on **repl.it**, where you can execute the code without having to install a Python environment, and you can make changes to my code or write your own programs (but a free login is required if you want to save them).

Thanks to everyone who has contributed to the site in 2017, either by adding their own solutions (programmatic or analytical), insights or questions, or by helping me source puzzles from back-issues of *New Scientist*.

1 September 2017

Posted by on **From New Scientist #1562, 28th May 1987** [link]

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

moretriangles 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!

[enigma412]

8 May 2017

Posted by on **From New Scientist #2272, 6th January 2001** [link]

Albion, Borough, City, Rangers and United played a tournament in which each team played each of the other teams once. Two matches took place in each of five weeks, each team having one week without a match.

One point was awarded for winning in the first week, 2 points for winning in the second week, 3 points for winning in the third week, 4 points for winning in the fourth week and 5 points for winning in the fifth week. For a drawn match each team gained half the points it would have gained for winning it. At any stage, teams that had gained the same number of points were regarded as tying.

After the first week A led, with B tying for second place. After the second week B led, with C tying for second place. After the third week C led, with R tying for second place. After the fourth week R led, with U tying for second place. After the fifth week U had won the tournament with more points than any of the other teams.

(1) Which team or teams finished in second place after the fifth week?

(2) Give the results of Albion’s matches, listing them in the order in which they were played and naming the opponents in each match.

This completes the archive of *Enigma* puzzles from 2001. There are now 1065 *Enigma* puzzles on the site, the archive is complete from the beginning of *Enigma* in February 1979 to January 1987, and from January 2001 to the final *Enigma* puzzle in December 2013. Altogether there are currently 59.5% of all *Enigmas* published available on the site, which leaves 726 *Enigmas* between 1987 and 2000 left to publish.

[enigma1116]

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