Enigmatic Code

Programming Enigma Puzzles

Tag Archives: by: Keith Austin

Enigma 1101: Disappearing numbers

From New Scientist #2257, 23rd September 2000

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}

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.


Enigma 1105: Road ants

From New Scientist #2261, 21st October 2000

Take a large sheet of paper and a black pen and draw a rectangle ABCD with AB = 10 metres and BC = 2 metres. Now draw lines to divide your rectangle into small squares, each of side 1 centimetre. Place your diagram so that A is due north of D and B is east of A. In each small square draw the diagonal that goes from northwest to southeast. Let P and Q be the mid-points of AD and BC, respectively. Then there is a black line PQ; remove it and replace it by a red line.

Amber is a small ant who can walk along the black lines in your diagram. North of PQ she covers a centimetre in 1 minute, but south of PQ she can cover a centimetre in 30 seconds. She is to walk from C to A and she chooses the quickest route.

1. How long does Amber take on her journey? Give the time, to the nearest second, in hours, minutes and seconds.

Ben is another ant who walks along the black lines. North of PQ he goes at the same speed as Amber, but not south of PQ. The fastest time for Ben to get from C to A is 24 hours.

2. South of PQ, how long does Ben take to cover a centimetre? Give the time, to the nearest second, in minutes and seconds.


Enigma 406: The ritual

From New Scientist #1556, 16th April 1987 [link]

I entered the jungle clearing and found, at the centre, six stones numbered 1, 2, 3, 4, 5, 6. My native guide demonstrated the ancient ritual which was enacted on that site.

First I had to arrange the stones in any order I wished, so I put them as 421365. I then noted the number on the first stone — all counting is from the left — is was 4, and so I picked up the fourth stone and placed it at the right hand end. I obtained 421653. Next, I looked at the second stone, a 2, and moved the second stone to the end, to give 416532. I repeated the procedure for the third stone to give 416532, and then for the fourth, fifth and sixth stones, to give successively, 416523. 465231, 652314. The final arrangement was called the result of the ritual.

Just then the high priestess, Sarannah, entered. She arranged the stones in a certain order, carried out the ritual and obtained the result 314625. My guide explained that the arrangement Sarannah started with was the result of applying the ritual to a very special arrangement, which she could not tell me.

What was the arrangement that Sarannah started with?


Enigma 402: A DIY puzzle

From New Scientist #1552, 19th March 1987 [link]

In the puzzle below, select some of the guest lists and discard the remainder. The lists you keep should make a puzzle which has exactly one solution and involves no more lists than is necessary.

Who did it?

by …

There has been a series of robberies at house parties recently. Each was clearly a one-man job — the same man each time — and each was an inside job. The possible suspects are Alan, Bryan, Chris, David, Eric, Fred, George, Harry, Ian, Jack, Ken and Len. The male guest list at the parties were as follows:

1. All but David, George and Len.
2. Bryan, Chris, Eric, Harry, Ian and Ken.
3. All but Bryan and Ken.
4. Chris and Ian.
5. All but Alan, Fred, Jack and Len.
6. Bryan, Chris, Ian and Ken.
7. All but Eric and Harry.
8. All but David and George.
9. All but Chris and Ian.
10. All but Alan and Fred.

Who carried out the robberies?

Which lists should you use in your puzzle?

What is the answer to your puzzle?


Enigma 1110: Dots and lines

From New Scientist #2266, 25th November 2000 [link]

Matthew and Ben are playing a game. The board is a 1-kilometre square divided into 1-centimetre squares. The centre of each small square is marked by a red dot.

Matthew begins the game by choosing a number. Ben then selects that number of red dots. Finally Matthew chooses two of Ben’s selected dots and draws a straight line from one to the other. Matthew wins if his line passes through a red dot other than those at its ends; otherwise Ben wins.

What is the smallest number that Matthew can choose to be certain of winning?

In the magazine this puzzle was incorrectly labelled Enigma 1104.


Enigma 398: Down on the farm

From New Scientist #1548, 19th February 1987 [link]

Farmer O. R. Midear has crossed a turnip with a mangel to get a tungel, and he now has many fields of tungels. As a tungel expert, he looks at a tungel to see if it is red or not, if it is smooth or not, and if it is firm or not. He knows that if a tungel is red then it is smooth.

Regulations have just been introduced which put fields of tungels into classes A, B, C, D, E; a field may be in more than one class.

Class A: All fields containing no red tungel.
Class B: All fields containing no smooth tungel.
Class C: All fields in which every red tungel is firm.
Class D: All fields in which every smooth tungel is firm.
Class E: All fields containing a red tungel which is not firm.

To test Farmer Midear’s understanding of the regulations he was asked to say which of the following statements are true.

1. If a field is in A then it is in B.
2. If a field is in B then it is in A.
3. If a field is in C then it is in B.
4. If a field is in B then it is in C.
5. If a field is in C then it is in D.
6. If a field is in D then it is in C.
7. If a field is in D then it is not in E.
8. If a field is not in E then it is in C.
9. In every field there is a tungel such that if it is not red then the field is in A.

What should Farmer Midear’s answer be?


Enigma 1114: Christmas changes

From New Scientist #2270, 23rd December 2000 [link]

Christmas is said to change things and so this enigma is an old puzzle with some changes.

Problem: You have to assign a digit to each of the 10 letters in the sum here:

When you have decided on an assignment of digits to the 10 letters, then your assignment is a solution of the problem if it satisfies at least one of the following conditions:

• At least one digit is assigned to more than one letter.
• When the digits are put into the addition sum it is not correct.
• The digit assigned to “U” is smaller than the digit assigned to “H”.

You need to find an assignment of digits to the 10 letters which is NOT a solution of the problem.

What is the value of BLEAT in your assignment?


Enigma 393: Decode the sum

From New Scientist #1543, 15th January 1987 [link]

In the following addition sum, different letters stand for different digits and the same letter stands for the same digit throughout.

Decode the sum.


Enigma 1121: Families

From New Scientist #2277, 10th February 2001 [link]

There are six families, each consisting of mother, father and child. The mothers are Amber, Barbara, Christine, Dorothy, Ellen and Frances; the fathers are George, Harry, Inderjit, James, Kenneth and Lewis; the children are Matthew, Naomi, Oliver, Peter, Quentin and Rachel. The other day, everyone kept a diary of who they met in the 24-hour period; of course, everyone met the other two members of their family, but they also met other people. These are the diary records, given by initials:

A met G, J, L, M, N, P;
B met H, J, K, O, P, R;
C met I, K, L, M, O, Q;
D met G, I, L, P, Q, R;
E met H, I, K, M, N, R;
F met G, H, J, N, O, Q.


G met M, N, P;
H met O, P, R;
I met M, O, Q;
J met N, O, Q;
K met M, N, R;
L met P, Q, R.

If I told you who the wife of Inderjit is, then you could not work out who the father of Oliver is.

Question 1: Who is the wife of Inderjit?

If I told you who the mothers of Oliver and Quentin are, then you could work out who the mother of Peter is.

Question 2: Who are the mothers of Oliver, Quentin and Peter?


Enigma 390: Which statements are false?

From New Scientist #1539, 18th December 1986 [link]

Each of the following six statements is true or false or we cannot say whether it is true or false.

(1) Either 2 or 3 is the first true statement in the list of six.
(2) We can say both 4 and 5 are true.
(3) 6 is false and/or 4 is true.
(4) 1 is true and/or 6 is true.
(5) 3 is false and/or 1 is true.
(6) Both 2 and 5 are true.

Which of the six statements are false?


Enigma 386: Triangle of stones

From New Scientist #1535, 20th November 1986 [link]

I emerged from the impenetrable jungle into a clearing, at the centre of which were 21 stones laid on the ground to form a triangle.

Enigma 386

Just then, three native girls approached the stones; each wore a coloured sarong, one red, one blue, one white. They painted the six stones in the bottom row, white, white, blue, blue, red, red, in that order from left to right, and placed a coconut on the single stone in the top row. My guide explained that this was a traditional game. The girls would go and turn in the order red, white, blue, red, white. At each turn the girl would move the coconut to a stone which touched the coconut’s present stone and which was on a lower row. The game ended when the coconut reached the bottom row, and the colour of its final stone indicate the winner.

My guide knew the girls and said that if red could not win she would try to help white to win, similarly white would help blue and blue would help red: all the girls knew this.

After the game they exchanged the colours on two of the stones and played again. Blue won the second game.

Who won the first game and what was the row of colours for the second game?


Enigma 1127: Lights out

From New Scientist #2283, 24th March 2001 [link]

There are 13 lights, A, B, C, …, L, M, in the dormitory and each one has its own switch. To save matron having to operate all 13 switches, 34 pairs of lights are connected. They are:


Whenever a switch is operated it changes its own light and each of the lights connected to it from on to off, or from off to on. When matron enters the dormitory at 9.00 pm all the lights are on. As an example, if she operates switch A then lights A, B, C, D and J go off. If she then operates switch J then lights A, D and J come back on and lights E and F go off.

Question: Which switches should matron operate so that all the lights are off when she has finished.

See also Enigma 1137.


Enigma 381: Island airlines

From New Scientist #1530, 16th October 1986 [link]

Come with us now to those 10 Pacific islands with the quaint native names A, B, C, D, E, F, G, H, I, J. These are served by Lamour and Sarong airlines. Each morning one plane from each airline leaves each island bound for another of the islands. No two planes from the same airline are going to the same island and the two planes leaving any island go to different destinations.

The planes all carry out the return journeys overnight and everything is repeated the next day and so on.

Each island has a beautiful queen. One morning, each queen left her island on the Lamour plane, staying overnight on the island she reached, and left the next morning on the Sarong plane. At the end of their two-day journey the queens of A, B, C, D, E, F, G, H, I, J found themselves on the islands of C, F, B, H, J, D, E, I, G, A, respectively.

Some time later, the queens made similar journeys, again starting from their home islands, but this time taking the Sarong planes on the first day and the Lamour planes on the second day. This time the Queens of A, B, C, D, E, F, G, H, I, J ended up on the islands of J, A, B, I, F, H, E, C, G, D respectively.

As the sun slowly sits in the west we say a fond farewell to a £10 book token, which will be sent to the first person to tell as the details of the 10 Lamour routes.


Enigma 1131: Numbers in boxes

From New Scientist #2287, 21st April 2001 [link]

I have a row of boxes numbered 1, 2, 3, …, in order, going on forever. Each box contains a piece of paper on which is written a positive fraction, for example box 1 contains 2/5. When I looked at the numbers in all the boxes I found the following was true:

If you chose any positive fraction then I can find a particular box so that all the numbers in the boxes after that particular box will be less than your fraction.

For example, if you choose 1/3 then all the numbers in the boxes after box 44 are less than 1/3.

This morning I was looking for 3 boxes containing numbers adding up to 1. In fact I made a list, L, of all such three’s of boxes.

Question: Which of the following statements can you say for certain are true?

(a) All the boxes on list L come before box 100;
(b) All the boxes on list L come before box 1,000,000;
(c) I can find a particular box so that all the boxes on list L come before my particular box.


Enigma 377: Cricket, lovely cricket

From New Scientist #1526, 18th September 1986 [link]

Enigma 377

Three runs were scored in each over and they were all scored in singles.

What was the number of runs scored at the fall of each wicket?


Enigma 1137: On, off, on, off

From New Scientist #2293, 2nd June 2001 [link]

There are 8 lights, A, B, C, D, E, F, G, H, in the junior dormitory and each one has its own switch. To save matron having to operate all 8 switches, 12 pairs of lights are connected. They are AB, AD, BC, BE, CF, CH, DF, DG, EF, EG, EH, GH.

Whenever a switch is operated it changes its own light, and each of the lights connected to it, from on to off or from off to on. When matron enters the dormitory at 9.00 pm, lights B, C, E and G are on and the other four lights are off. As an example, if she operates switch F then lights D and F come on and lights C and E go off. If she then operates switch E the lights E and H come on and lights B, F and G go off.

Question 1. Is it possible for matron to operate certain switches so that, when she has finished, all the lights are off? If it is possible, which switches should she operate?

If it is not possible, which one of the four lights that were off at 9.00 pm should additionally have been on at 9.00 pm so that matron could have operated certain switches and finished with all the lights off?

The situation is similar in the senior dormitory except that there are 100 lights and 400 pairs of lights are connected and all the lights are on at 9.00 pm.

Question 2. Can we say for certain that matron can find a selection of switches to operate so that when she has finished all the lights are off?


Enigma 373: Date the painting

From New Scientist #1522, 21st August 1986 [link]

Enigma 373

(R = Red, B = Blue, G = Green, Y = Yellow)

You will no doubt have recognised this as the latest masterpiece by the artist Pussicato, which has just gone on display at the National Gallery. I visited Pussicato in his studio at the end of March, just before he began work on the picture. He showed me the canvas divided into 36 squares which were numbered 1 to 36. He planned to paint the squares in order, one per day starting with square 1 on 1 April, using the four colours in strict rotation. He had numbered the squares so that from one day to the next he always moved to an adjacent square either horizontally or vertically; also, squares 1 and 36 were similarly adjacent. As I looked at the canvas, I pointed to two horizontally adjacent squares and remarked that the right-hand one contained a number which was 8 times the number in the left-hand square. Pussicato had no reply.

On what date did Pussicato paint the top left hand square of the picture?


Enigma 1143: Count and count

From New Scientist #2299, 14th July 2001 [link]

You will need a large sheet of lined paper. Draw 9 vertical lines to divide the sheet into 10 columns. On the first line write the headings of the columns, 0, 1, 2, …, 8, 9. On the second line write your own choice of digits 0 to 9, one digit in each column, subject to the conditions that the digits are not all the same and they are not all different. Fill in line 3 as follows. Count how many 0’s there are in line 2 and put the answer on line 3 in the 0 column. Count how many 1’s there are in line 2 and put the answer on line 3 in the 1 column. Repeat for the other digits. In the same way that line 3 was filled in using line 2, so you fill in line 4 using line 3. Continue in this way filling in, in turn, lines 5, 6, 7, …

What are the possible lines of digits the might be on line 1000 of your sheet of paper?


Enigma 368: What’s the truth

From New Scientist #1517, 17th July 1986 [link]

Each of the following six statements is either true or false.

1. Statements 2 and 3 are both true or both false.
2. Exactly one of statements 4 and 5 is true.
3. Exactly one of statements 4 and 6 is true.
4. Exactly one of statements 1 and 6 is true.
5. Statements 1 and 3 are both true or both false.
6. Exactly one of statements 2 and 5 is true.

Which of the six statements are true?


Enigma 1147: Multiply and add

From New Scientist #2303, 11th August 2001 [link]

Matthew is practising his multiplication and addition. He had a board (as below) and nine cards labelled 1,2,3,…,8,9.

Enigma 1147

He shuffles the cards and deals them on to the board, one to each square. For each square, he multiplies the number on the board by the number on the card in that square. Finally he adds together the nine products he has obtained to give his final number.

When I watched Matthew play the game his final number was 545. When he had finished he then moved some of the cards. Each card he moved went one square horizontally or vertically, but not diagonally, to another square on the board. I can’t remember all the cards he moved, but I do know he moved the cards in the four corner squares. He then had a new layout with one card in each square and he repeated his multiply and add routine. When he told me his final number, I asked him how many factors it had. After a while he told me he had found 20 but had still not finished his search.

What was Matthew’s final number after he had moved the cards?