Enigmatic Code

Programming Enigma Puzzles

Category Archives: enigma

Enigma 504: Hooray for Hollywood

From New Scientist #1656, 18th March 1989 [link]

Twentieth Century Lion Studios has just held a week-long film festival celebrating 60 years of talking pictures. It first selected seven of the studio’s legendary stars. It then chose seven of the studio’s classic films so that each of the 21 possible pairs of stars appeared together in one of the films.

The stars were selected from Fred Astride, Humphrey Bigheart, Joan Crowbar, Bette Daybreak, Judy Garage, Clark Gatepost, Katherine Hipbone, Barbara Standup, James Student, Spencer Treacle, John Weighing. The films were chosen from the following list in which each film is given with its stars:

Cosiblanket, JC, JG, BS
Top Hit, BD, JG, KH
Stagecrouch, HB, CG, KH
A Star is Bone, HB, JC, JS
Mildred Purse, CG, ST, JW
High None, HB, JG, ST
King Koala, FA, JG, CG
Random Harpist, BD, KH, JS
Now Forager, JC, BD, CG
Mrs Minimum, CG, JS, JW
The Adventures of Robin Hoop, KH, BS, JW
The Maltese Foghorn, FA, JC, ST
Mr Deeds goes to Tune, BS, JS, ST
Meet me in St Lucy, BD, CG, BS
Gone with the Wine, FA, HB, BD
Singing in the Rind, FA, JC, KH
Mutiny on the Bunting, BD, ST, JW
The Best Years of our Lifts, FA, HB, BS
Double Identity, FA, JG, JS

Which seven films were selected?


Enigma 1006: Booklist

From New Scientist #2161, 21st November 1998

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?


Enigma 503: Early in the season

From New Scientist #1655, 11th March 1989 [link]

Five football teams are to play each other once. After some of the matches had been played — in fact no team had played more than two matches — some details of the situation looked like this:

Enigma 503

(Two points are given for a win and one point to each side for a draw).

Find the score in all the matches that had been played.


Enigma 1007: Triangular and Fibonacci cubes

From New Scientist #2162, 28th November 1998

Harry, Tom and I were trying to find a 3-digit triangular number, a 3-digit Fibonacci number and a 3-digit perfect cube that between them used 9 different digits. (Triangular numbers are those that fit the formula n(n+1)/2; in the Fibonacci sequence the first two terms are 1 and 1, and every succeeding term is the sum of the previous two terms).

We each found a valid solution. Mine had one number in common with Harry’s solution and a different number in common with Tom’s solution; otherwise, the numbers used in our solutions were all different.

List in ascending order the numbers used in my solution.


Enigma 502: Fill this up!

From New Scientist #1654, 4th March 1989 [link]

Here is a partly filled-in Magic Square in which I have replaced digits with letters, different letters being used consistently for different letters.

In the complete Magic Square the sum of the three numbers in each row, the sum of the three numbers in each column and the sum of the three numbers in each long diagonal is the same, namely PUT.

Enigma 502

Please fill this up with numbers (put a number in every square).


Enigma 1008: Triangles and squares

From New Scientist #2163, 5th December 1998

I have a solid irregular tetrahedron. Its four triangular faces are coloured red, yellow, blue and green, with the green face having the largest perimeter.

The lengths of the six edges are all different and each equals a number of centimetres which is a perfect square. For example, one of the edges is 16 centimetres long.

What are the lengths of the three sides of the green face?


Enigma 501: A reciprocal arrangement

From New Scientist #1653, 25th February 1989 [link]

“As you insist on disturbing my peace of mind with puzzles”, remarked Potter to Kugelbaum as they sat down to drinks at the Maths Club, “it is only fair that you submit to the same fate”. Kugelbaum agreed. “The Egyptians expressed fractions as sums of reciprocals”, continued Potter. “For example, they wrote 3/8 = 1/8 + 1/4. That and the notion of getting one over you inspires this puzzle:”

“The smallest integer, U, such that 1/U may be expressed as the sum of exactly two reciprocals in exactly and only two distinct ways is 2; for 1/2 = 1/4 + 1/4 and 1/2 = 1/3 + 1/6 and there are no other ways of doing it. The second smallest is 3, since 1/3 may be expressed as the sum of exactly two reciprocals in exactly and only two distinct ways: 1/3 = 1/6 + 1/6 and 1/3 = 1/4 + 1/12. Again there are no other ways. But the third smallest value of U is not 4, since 1/4 may be expressed as the sum of two reciprocals in three distinct ways.”

“Yes”, said Kugelbaum in reply, “and namely 1/8 + 1/8; 1/6 + 1/12; and, of course, 1/5 + 1/20. What is your question?”

Potter drew himself up: “What is the eighth smallest value of U, such that 1/U is expressible as the sum of exactly and only two reciprocals in exactly and only eight distinct ways?”

Kugelbaum’s eyes glazed over and the cogs began to whir. In fact, Potter didn’t even know if the question had an answer and so when Kugelbaum gave the answer, he had to take it on trust. Given that Kugelbaum is never wrong, what was his answer?


Enigma 1009: Squared square

From New Scientist #2164, 12th December 1998

The diagram shows the simplest solution to the classical problem of dissecting a square into a number of smaller squares all with sides which are integers, no two the same. Unfortunately, the dimensions (several of which are prime numbers) have been deleted.

By studying the diagram with care can you determine the side of the outer square?


Enigma 500: Child’s play

From New Scientist #1652, 18th February 1989 [link]

The children at the village school have a number game they play. A child begins by writing a list of numbers across the page, with just one condition, that no number in the list may be bigger than the number of numbers in the list. The rest of the game involves writing a second list of numbers underneath the first; this is done in the following way. Look at the first number — that is, the left-hand one, as we always count from the left. Say it is 6, then find the sixth number in the list — counting from the left — and write that number in the first place in the second row — so it will go below the 6. Repeat for the second number in the list, and so on. In the following example, the top row was written down, and then playing the game gave the bottom row:

6,  2,  2,  7,  1,  4, 10,  8,  4,  2,  1
4,  2,  2, 10,  6,  7,  2,  8,  7,  2,  6

The girls in the school use the game to decide which boys are their sweethearts. For example, Ann chose the list of numbers:

2,  3,  1,  5,  6,  4

For a boy to become Ann’s sweetheart he has to write down a list of numbers, play the game, and end with Ann’s list on the bottom row.

Bea chose the list:

2,  3,  2,  1,  2

and Cath the list:

3,  4,  5,  6,  7,  1,  2,  5,  7,  3,  6,  9

Find all the lists, if any, which enable a boy to become the sweetheart of Ann, of Bea, and of Cath.

Enigma 1736 is also called “Child’s play”.


Enigma 1010: Christmas list

From New Scientist #2165, 19th December 1998

As usual, this Christmas I shall be given a diary. So I shall have to transfer information from my current diary into it. This includes the dates of 11 birthdays which I have to remember, no two of which are in the same month.

If you write each of these different dates as a number (so that, for example, New Year’s Day would become 11, and Christmas Day would become 2512) and write the 11 numbers in increasing order, then they have a surprising property. The difference between the first and the second is the same as the difference between the second and the third, which is the same as the difference between the third and the fourth, and so on through the list. Also no two adjacent numbers arise from birthdays in consecutive months. More of the 11 birthdays will fall on Saturdays in 1999 than did in 1998.

What are the dates of the Saturday birthdays in 1999?


Enigma 499: Mathematical spelling test

From New Scientist #1651, 11th February 1989 [link]

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

Enigma 499

Rewrite the division sum with the letters replaced by digits.


Enigma 1011: The ribbon’s reach

From New Scientist #2165, 19th December 1998

Mary is wrapping her last Christmas present, which is a rectangular box which measures 1 metre by 1 metre by 2 metres. She has attached one end of a piece of ribbon to a corner of the box. Amazingly, she finds that the ribbon is just long enough to reach any point on the surface of the box; however if it were any shorter it would not be able to do that.

How long, to the nearest millimetre, is the ribbon?


Enigma 498: Vowel count

From New Scientist #1650, 4th February 1989 [link]

Vowel count
by Susan Denham

In this Enigma (from “vowel” above to “?” at the end) if you put in the right numbers (in words) in the spaces then there will be ……. a’s, …… e’s, ……. i’s, …… o’s and …… u’s.

After the correct things are put in all the spaces, how many vowels will there be (“vowel” – very end “?” inclusive)?

Note: I have reproduced the text of the puzzle verbatim.


Enigma 1012: Pieces of eight

From New Scientist #2168, 9th January 1999

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.


Enigma 497: The longest puzzle in the world?

From New Scientist #1649, 28th January 1989 [link]

Hannah, Joan and Sarah each live at a different one of the three houses numbered 1, 2, 3. Alan, Michael and Peter each painted a different one of those three houses. Each of the three women met a different one of the three men at the shops.

There are four clues as to who lives where, and so on. However, as the clues are very long they are given here only in a condensed form, and so you may wish first to write them out in full, as indicated.

(1) Write out “the man who painted the house of the woman who met, at the shops,” (1988 + the number of the house of the woman met by Peter) times. Write the copies of the repeated phrase one after the other, and then, in front of the first copy write “Alan is” and, after the last copy, write “Alan.” to make one very long sentence which is the first clue.

(2) This is similar to clue (1), but has “Michael is” at the start and “Peter.” at the end.

(3) Write out “the woman who met, at the shops, the man who painted the house of” (1066 + the number of the house painted by the man who met Hannah) times. Add “Hannah is” at the start and “Joan.” at the end.

(4) This is similar to clue (3), but the repeated phrase is “the house that was painted by the man who met, at the shops, the woman who lives at”, the clue starts with “Number 1 is” and end with “Number 2.”

What are the facts, that is, who lives where, who painted which house, who met who?


Enigma 1013: Party trick

From New Scientist #2169, 16th January 1999

I have a numerical party trick which relies on the fact that my telephone number is a rather special four-digit one.

I ask a volunteer to choose any two odd or two even digits, not necessarily different as long as they’re not both zero. I then ask him or her to make up a six-figure number as follows: use one of your chosen digits, which is not zero, in the first and last place of the six-figure number; use the other chosen digit in the third and fourth places; and use the average of the two chosen digits in the second and fifth places.

Having constructed the number in this way, I am able to announce that it is divisible by the four-figure number which is my telephone number. It always works out like that.

When I tried this on my nephew he chose his two digits, constructed the number, entered it in his calculator and divided by my telephone number. The whole number answer to his division sum turned out to be a factor of my telephone number.

What was the six-figure number which he constructed?


Enigma 496: A princely sum

From New Scientist #1648, 21st January 1989 [link]

In the following addition sum, each letter stands for a different digit. Replace the letters with digits.

Enigma 496

Enigma 420 is also called “A princely sum”.


Enigma 1014: Mirror image

From New Scientist #2170, 23rd January 1999 [link]

Harry was playing about with his calculator and keyed in a 4-digit number. He placed a mirror behind and parallel to the display, and added the reflected number, which was smaller, to the number on the display. This gave him a 5-digit sum.

He then again keyed in the original number, and this time subtracted the reflection from it.

He divided the sum by the difference and found that the quotient was a 4-digit prime.

What was his original number?


Enigma 495: Lack of details

From New Scientist #1647, 14th January 1989 [link]

Four football teams are to play each other once. After some of the matches had been played a document giving a few details of the matches played, won, lost and so on was found. This time I am glad to say that, although it was rather a mess, all the figures given were correct. Here it is:

Enigma 495

(Two points are given for a win and one point to each side in a drawn match).

Find the score in each match.


Enigma 1015: Money-spinner

From New Scientist #2171, 30th January 1999 [link]

In my local pub there is an electronic “slot machine” which offers a choice of various games. In one of them, called Primetime, after inserting your pound coin the 9 digits 1-9 appear in random order around a circle. Then an arrow spins and stops between two of the digits. You win the jackpot if the two-digit number formed clockwise by the two digits on either side of the arrow has a two-figure prime factor. So, if the digits and arrow ended up as above, you would win the jackpot because 23 is a factor of 92.

However, with the digits in the same position but with the arrow between 8 and 1 you wound not win.

I recently played the game. The digits appeared and the arrow started to spin. But I realised to my annoyance that, no matter where the arrow stopped, I could not with the jackpot.

Starting with 1, what is the clockwise order of the digits?


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