Enigmatic Code

Programming Enigma Puzzles

Tantalizer 489: Buttons and bows

From New Scientist #1040, 24th February 1977 [link]

Great is the rejoicing in the firm of Furbelows over the engagement of Bertha Button of the button department to Bertie Bow, beau of the bows. Since Miss button is the fanciest of the three spinsters in buttons, while Mr Bow is quite the most eligible of the eight bachelors in the bows, it may seem none too astonishing that Cupid has singled them out. But, considering the number of bachelors in buttons and of spinsters in bows, it is as well that the merry archer does not loose his shaft at random. For, had he done so, the chances are 29 to 23 in favour of an engagement between two members of the same department.

How many bachelors are there in buttons?


Puzzle 80: Addition: letters for digits

From New Scientist #1131, 30th November 1978 [link]

Below is an addition sum with letters substituted for digits. The same letter stands for the same digit wherever it appears, and different letters stand for different digits:


Write the sum out with numbers substituted for letters.


Enigma 1127: Lights out

From New Scientist #2283, 24th March 2001

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 384: Hang it!

From New Scientist #1533, 6th November 1986 [link]

Our local club’s darts’ champion Rice Robswit was about to throw three darts in an attempt to win the match. The score he needed could have been got with one treble, but in order to show his prowess and to finish with a double he went for a single, a treble of a different number, and a double of a different number again (avoiding the bulls) in order to give him exactly the total he wanted. Being an experienced player he did not actually look at the numbers around the board — he simply threw the three darts into exactly the positions he had planned.

The crowd roared their approval until someone pointed out that, after the board had fallen on the floor at the end of the previous player’s throw, it had been hung up upside down. There was a groan from the crowd until, on a fresh and proper count, it was found that Rice had still scored his correct required total.

What was that total?

(The numbers around a darts board are in the order 20, 1, 18, 4, 13, 6, 10, 15, 2, 17, 3, 19, 7, 16, 8, 11, 14, 9, 12, 5, 20, …).


Tantalizer 490: Diplomatic niceties

From New Scientist #1041, 3rd March 1977 [link]

Before assuming office as governor of Coconut Island, Sir Donald Duck briefed himself as best he could. There were, he discovered, four chiefs called Fe, Fi, Fo and Fum. The mark of chiefly rank was a turkey feather, red or green at will. The senior chief wore an old pair of Wellington boots and the others went barefoot. Fe always spoke the truth, Fi never, Fo pleased himself and Fum spoke the truth when and only when wearing a green feather.

Knowing no more than this, Sir Donald landed with pomp and found the four chiefs awaiting him. He shook hands all round and inquired, “What is the name of the senior chief?” One chief replied “Fe”, another “not Fum” and a third “Fo”. Sir Donald did not hear the fourth reply but it did not matter, since, being a Balliol man and so very clever, he worked out the name of the senior chief without it.

What was the name of the senior chief?


Puzzle 81: Uncle Bungle and the vertical tear

From New Scientist #1132, 7th December 1978 [link]

It was, I’m afraid, typical of Uncle Bungle that he should have torn up the sheet of paper which gave particulars of the numbers of matches played, won, lost, drawn and so on of four local football teams who were eventually going to play each other once. Not only had he torn it up, but he had also thrown away more than half of it onto, I suspect, the fire, which seems to burn eternally in Uncle Bungle’s grate. The tear was a vertical one and the only things that were left were the “goals against” and the “points” — or rather most of the points, for those of the fourth team had also been torn off.

What was left was as follows:


(2 points are given for a win and 1 for a draw).

It will not surprise those who know my uncle to hear that one of the figures was wrong, but fortunately it was only one out (i.e. one more or one less than the correct figure).

Each side played at least one game, and not more than seven goals were scored in any match.

Calling the teams ABC and D in that order, find the score in each match.


Enigma 1128: Daffodils

From New Scientist #2284, 31st March 2001

Saul Tregenza is a market gardener. He has a field in which he decided that it could be profitable to plant equal rows of daffodils. On a whim prompted by helping with his daughter’s homework, he decided that the number of rows and the number of bulbs in each row should be prime, and that all the digits that would form the two numbers and their product would be different.

Luckily, he found that the field is long enough to hold the maximum possible number of plants under these whimsical conditions.

What was the maximum possible number of bulbs he could plant?


Enigma 383: Stop watch

From New Scientist #1532, 30th October 1986 [link]

My watch is pretty accurate — when it’s working. It has a sweep second-hand that advances in jerks, one each second; if the watch stops, it stops with the second hand pointing directly to one of the 60 “second” (or “minute”) marks round the rim. At noon all three hands are (naturally) pointing directly to the zero-minute mark.

At noon yesterday my watch was certainly working, but when I checked it at 6:00 pm I found that it had stopped. The angles the three hands made with each other all differed from 120 degrees by less than 1 degree.

At what time had my watch stopped? (Hours : minutes : seconds).


Tantalizer 491: Cats and dogs

From New Scientist #1042, 10th March 1977 [link]

Six proud but ill-acquainted owners were to be heard exchanging remarks at the village pet show. I noted down some of them and off you a brief selection:

Amble to Bumble: “Dimwit keeps a dog”.

Bumble to Crumble: “Egghead and Fumble have pets of the same sort”.

Crumble to Amble: “Bumble’s pet is not the same sort as yours”.

Dimwit to Egghead: “Bumble has the smelliest dog in the village”.

Egghead to Bumble: “Crumble keeps a dog”.

Fumble to Dimwit: “Crumble’s pet is not the same sort as mine.”

Each man has a cat or a dog (not both) and has spoken the truth if and only if addressing someone with the same sort of pet. “Same sort” means merely cat or dog — finer distinctions, such as that between collie and corgi, do not count.

Who owns what?


Enigma 1129: Minimal change

From New Scientist #2285, 7th April 2001

The coins from 1p to 100p in circulation are for 1p, 2p, 5p, 10p, 20p, 50p and 100p. If one receives 40p in change for a purchase it can be paid in a minimum of 2 coins (20p + 20p) but could be paid in one coin more than that minimum number (20p + 10p + 10p). But a few amounts of change cannot be paid for in one coin more than the minimum possible number of coins, for instance 1p and 50p.

Harry, Tom and I each paid 100p for an item priced such that the change due could not be paid in one coin more than the minimum possible number of coins. Each item cost a different amount.

Next day Harry and Tom bought the same items at the same price as on the previous day, whereas I bought a different item at a different price from any of the others, which again was such that the change due from 100p could not be paid in one coin more than the minimum possible number of coins. Each day the total cost of the three purchases was a prime number of pence.

How much did each of my purchases cost?


Enigma 382: Dice

From New Scientist #1531, 23rd October 1986 [link]

While rummaging through the library stacks I found a Latin manuscript which translates roughly as follows:

Dear Uncle,

The natives in these desert parts use tetrahedral dice, rather than the cubical ones of Rome; a tetrahedral die sticks into the sand point first, which is useful when there are no pavements to play on. Each face on a given die is numbered with a different natural number (being nomads they count zero as a natural number). A set consists of two dice numbered in such a way that no score with a single die or with a set is a power of two, apart from the highest score possible with a pair. (In case you are rusty, Uncle, 1 is counted as a power of 2). What’s more, you can’t throw a total of zero with a set. Every other score up to the highest can be thrown with a pair.

I won’t bore you with an account of my losses, but I am told that the information given is sufficient to determine the numbers on each of the two dice.



What numbers are inscribed on the two dice?


Puzzle 82: A cross number

From New Scientist #1133, 14th December 1978 [link]


(There are no 0’s).


1. Each digit is odd and is greater than the one before.
4. The digits are all different and this is a multiple of the number which is 3 greater than 1 down. Even when reversed.
5. A perfect cube.


1. 17 goes into this.
2. A multiple of 1 down.
3. Each digit is odd and is less than the one before.

One clue is incorrect. Which one?

With which digit should each square be filled?


Enigma 1130: Time and again

From New Scientist #2286, 14th April 2001

To practise my long multiplication I have taken two three-figure numbers (using six different digits between them) and multiplied the first by the second. Then, as a check, I multiplied the second by the first. The results are shown with dashes for all non-zero digits:


What were the two three figure numbers?


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.


Tantalizer 492: Bon appetit

From New Scientist #1043, 17th March 1977 [link]

If you ever take a holiday on the little island of Mandible, be sure to sample the local food. The basic element is a squash, called a Tiddly, which sells at KL francs per portion. One of these together with a Widdly and an Om make a satisfying meal for LJ francs. But you do not have to have a Tiddly every time and there is much to be said for having just the Widdly and the Om, in which case the dish will cost JL francs. Yet, the Widdly being a bug-eyed lizard and the Om a fried roll filled with peppered ants, you might do well to order a Pom too, thus raising the cost from JL to KM francs. Finally there is the famous Mandible Monster, which consists quite simply of Tiddly, Widdly, Om and Pom and costs MK francs.

Mandible money is straightforward so I have tried to confuse you by replacing digits with letters. Thus JK means 10×J + K and so on.

My own favourite dish is the Tiddly Om Pom (which I had previously supposed to be the French for a drunken man and an apple). In plain digits, what does it cost, given that a Widdly costs J francs more than an Om?


Puzzle 83: Division: some letters for digits, some missing

From New Scientist #1134, 21st December 1978 [link]

In the following division sum, some of the digits are missing, and some are replaced by letters. The same letter stands for the same digit wherever it appears.

Puzzle 83

Find the correct sum.


Enigma 1131: Numbers in boxes

From New Scientist #2287, 21st April 2001

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 380: Answer what?

From New Scientist #1529, 9th October 1986 [link]

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


Write the sum out with the letters replaced by digits.


Tantalizer 493: Prize guys

From New Scientist #1044, 24th March 1977 [link]

I met Tom on his way back from the pet show looking pleased with himself. “Prize for my budgie, prize for my cat, prize for my dog”, he explained, “How did Dick and Harry do?”

“Don’t you know?” I asked.

“Not a thing.”

“They too got a prize each for each of a budgie, cat and dog. So you scooped all those nine prizes between you — no ties, incidentally. Dick’s lowest prize was for his budgie.”

“Oho,” exclaimed Tom after reflection, “so Harry did better with his cat than with his dog.”

He then listed all nine prizes correctly. Can you?


Enigma 1132: Phone back

From New Scientist #2288, 28th April 2001

The PIN code on George’s cash card is a semi-prime number, that is to say it is the product of two different prime numbers. He discovered some time ago that the PIN code multiplied by his car registration number gives his six-digit phone number, which does not begin with a zero. But George has now discovered a slightly more obscure coincidence. If he subtracts his house number from his phone number and multiplies the result by his house number, the result is his phone number with its digits in reverse order!

What is George’s car registration number?