# Enigmatic Code

Programming Enigma Puzzles

## Enigma 1293: Reverse Fahrenheit

From New Scientist #2451, 12th June 2004

“Multiplying by 9/5 and adding 32,” I explained to my clever nephew George, “is useless in practice. What you need is some memorable equivalents, like 10 °C being 50 Fahrenheit. Here’s one I’ve invented: 16 °C = 61 °F. See, to get from one to the other you just reverse the two digits.”

“Actually 16 °C = 60.8 °F,” I said.

“So 61 is near enough,” I said.

“Near enough is not exactly right.”

“But you cannot do it exactly,” I objected sourly.

“You can’t, because you insist on boring old base 10. But I bet I can, using other bases,” George retorted. Off he went to investigate, and was soon back. “-90 °C = -130 °F,” he said, “and to base 21 this says -46 °C = -64 °F. I have other examples, including two between the freezing and boiling points of water.”

What were the two examples that George found? Give your answers in the form x °C = y °F where x and y are written in base 10 (and x lies between 0 and 100).

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment. The current estimate is that the line will be connected at the end of September 2014.

## Enigma 222: Hard astern

From New Scientist #1368, 28th July 1983 [link]

Admirals Drake, Jellicoe, Nelson and Raleigh were lately piped aboard HMS Beefeater for purposes of splicing the mainbrace. Each arrived immaculate in his well-cut uniform, did his share of the work to the full and departed with dignity barely askew. But (sh!) Drake presently realised that he had somehow come to be wearing a pair of trousers too small for him and a similarly tight feeling presently crept over Jellico. Nelson and Raleigh too found themselves clad in the wrong breeches.

Each promptly dispatched the trousers in his possession to one of the others. Each received one pair, not sent by the admiral to whom he had sent one and, alas, not his own. Nelson received a pair too tight for him.

Well, shiver my timbers, they then did the same again. Each sent off the pair in question. Each received a pair, not from the Admiral he had fired one at, not one which he had had already and still not his own. Drake received a pair too large for him, his own being now lodged with either Jellicoe or Nelson.

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment. The current estimate is that the line will be connected at the end of September 2014.

## Enigma 1294: Think of an animal

From New Scientist #2452, 19th June 2004 [link]

My young Australian nephew is getting good at arithmetic, and is gradually learning the names of a good number of animals. So I set him the following puzzle:

Think of a number, multiply it by 7, add 35, divide by 7, then take away the number you first thought of. Letting 1=A, 2=B, 3=C and so on, work out which letter your answer represents. Now think of an animal beginning with that letter.

In advance I had written “elephant” on a piece of card to amaze him. However, after thinking of a number he then performed the next four numerical steps in the wrong order, getting a positive whole number at each stage. He then worked out which letter his answer represented. However, he was unable to think of an animal beginning with that letter.

What number did he first think of, and what letter did he end up with?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment. The current estimate is that the line will be connected at the end of September 2014.

## Enigma 221: Moidores

From New Scientist #1367, 21st July 1983 [link]

I wonder if you could answer the following question from Hind’s Algebra, “designed for the use of students in the University”, published at Cambridge in 1839.

“In how many different ways may £100 be paid in crowns and moidores?”

A. Am I to include ways using only crowns or only moidores?

B. How many shillings is a moidore worth?

The answer to A, I can tell you, is “no”. The answer to the question is 14.

Now can you answer B? You will find you cannot be certain. But, assuming a moidore to be worth an exact number of shillings, what is the smallest and what is the greatest possible number of shillings in a moidore?

(Note to overseas and young readers: £1 = 4 crowns = 20 shillings).

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment. The current estimate is that the line will be connected at the end of September 2014.

## Enigma 1295: United could win

From New Scientist #2453, 26th June 2004 [link]

Albion, Borough, City, Rangers and United are playing another tournament in which each team plays each of the other teams once. Two matches are played on each of five successive Saturdays, each of the five teams having one Saturday without a match.

Three points are awarded for a win and one point for a draw. The third Saturday’s matches have just been played. The current points table shows the teams standing in the order Albion, Borough, City, Rangers, United; some of these teams are tied on points and separated only by goal difference, but United have fewer points than any other team. But the United players know that if they win their last two matches their team is sure to end up with more points than any other team. Albion beat City. What are the results of the other five matches that have been played?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

From New Scientist #1366, 14th July 1983 [link]

Not another mad letters-for-digits puzzle? Yes, but even crazier than usual. As always, different letters stand for different digits, and the same letter stands for the same digit from beginning to end. But this powerful example is quite mad, for example: ENIGMA = MAD.

As a further prime example of silliness, may I tell you too that PRIME is prime?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 1296: ET

From New Scientist #2454, 3rd July 2004

George drew an equilateral triangle, 3 units each side, divided into unit grid triangles, and asked his son how many triangles of all possible integer dimensions he could see in the diagram.

“Correct,” said George, “I have now defined the ET function, which stands for Embedded Triangles.  For any positive integer the ET function is the number of equilateral triangles of all possible integer sizes and orientations which are formed by the grid lines in an equilateral triangle of the given length of side.  Hence ET(3) is 13.

George then drew several integer sided equilateral triangles of different sizes, each divided into unit grid triangles.

“Can you see anything interesting about these, son?”

“Yes, Dad. The ET function of the largest is the sum of the ET functions of the others.”

“Right again. And no smaller triangle can be the largest of such a group.”

What are the lengths of the sides of the triangles?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 219: Matchwork, patchwork

From New Scientist #1365, 7th July 1983 [link]

Five football teams — ABCD and E — are all to play each other once. After some of the matches had been played a table giving some details of the matches played, won lost etc, looked like this:

Find the score in each match so far played.

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 1297: In order to be true

From New Scientist #2455, 10th July 2004

We write “b + n = f + l + 1″ to stand for the sentence, “the number of b’s plus the number of n’s equals the number of f’s plus the number of l’s plus one.”

In a similar way, write each of the following as a sentence in words;

m + t = p + r + 2;
a + e = h + r + 3;
f + r = h + l + 4;
l + n = f + p + 5;
t + u = a + e + 6.

Your problem is to put the six sentences you have (including the first example) into a certain order so that you have a paragraph of text with the following property:

If you take the text from the start of the paragraph up to the end of any particular sentence and count the letters in that text then you will find that particular sentence is a true statement about the text.

List the final words of the six sentences in the order in which they occur in the paragraph.

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 218: Relatively speaking

From New Scientist #1364, 30th June 1983 [link]

The professor during his lecture on relativity asked: “If I am in a spacecraft travelling at half the speed of light and pass another craft travelling in the opposite direction at a quarter of the speed of light, what is our relative velocity?”

“Three quarters of the speed of light,” replied one student.

“You weren’t paying attention at my last lecture,” said the professor. “We proved that, according to the special theory of relativity, when two velocities are to be added then the result is not their sum but this,” he broke off to write $(v_1 + v_2) / (1 + v_1 v_2 / c^2)$ on the board then continued, “where c is the velocity of light — 300 000 kilometres per second.

“Is it possible for two equal rational velocities to be added so that the result is an integral number of 1000 kilometres (we shall say megametres) per second?”

“No professor,” answered a bright student. “But if the speed of light is decreased by an integral number of megametres per second then it is possible.”

“But you can’t reduce the speed of light! — It is constant,” protested the professor.

“But we can imagine it to be less,” persisted the student.

The professor then suggested the amount his student had taken as the velocity of light.

“I took it as more than that, professor.”

“In that case I calculate what you took as the speed of light and all the possible sums of the equal velocities.”

Can you?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 1298: Odd change

From New Scientist #2456, 17th July 2004

I have some coins in my purse whose total value is less than 1 pound sterling. I have tried to make various totals using one or more of these coins and I have discovered two interesting facts.

First, each possible total can only be achieved by one particular combination of denominations. Secondly, the number of different totals possible equals the total value of the coins in pence. If I added any number of 1 pound coins to my purse those two facts would still be true.

Current UK coins with a value less than 1 pound are 50p, 20p, 10p, 5p, 2p, and 1p.

How much money do I have in total in my purse?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 217: Auntie Greta’s age

From New Scientist #1363, 23rd June 1983 [link]

“You want to know how old I am?” said Auntie Greta to her two nieces. “Well, as you know, your ages (in years) have no common factor higher than 1. My age (in years) is not an exact multiple of the age of either of you, but the square of my age is exactly the average of the cubes of your ages. So I am …”

How many years old?

This puzzle brings the total number of Enigmas on the site to 700. There’s a continuous archive of puzzles from Enigma 1299 on 24th July 2004 until the final puzzle, Enigma 1780 at the end of 2013 (482 puzzles). There is also a continuous archive from the start of Enigma in February 1979, up to Enigma 217 on 23rd June 1983 (217 puzzles), along with a lone puzzle from 2000 this brings us to 700 Enigmas in total, and leaves around 1085 puzzles to source and upload.

In total just over 39% of all Enigma puzzles are now available on the site. I will continue to expand the archive as time and internet connection allow.

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 1299: Pipe dreams

From New Scientist #2457, 24th July 2004

A pipe manufacturer ships pipes of 3 different radii in the same, square-section, box. The pipes just touch each other and the box as shown.  If the middle-sized pipes have radii 4 centimetres what are the radii of the smallest pipes?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 216: Point to point

From New Scientist #1362, 16th June 1983 [link]

As a challenging geometry puzzle, I asked my son to mark a prescribed number of points on a piece of paper, no three of them being in a straight line, and then to join each of the points to each of the others by straight lines. I knew by my choice of the number of points that he would not be able to do this without at least two of these lines crossing.

But I asked him to do it with some of his lines drawn in red, and the rest drawn in blue, and in such a way that it would be impossible to find a red triangle or a blue triangle in the whole configuration. This he managed.

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 1300: Numeros triangulares

From New Scientist #2458, 31st July 2004

This puzzle has appeared as an Enigma puzzle in both English and French versions, but incredibly it also works in Portuguese.

Triangular numbers are those that fit the formula n×(n+1)/2, like 1, 3, 6 and 10. In the following statement digits have been consistently replaced by capital letters, different letters being used for different digits: UM, TRES, SEIS, DEZ are all triangular numbers, none of which starts with a zero.

Which numbers are represented (in this order) by UM, TRES, SEIS, and DEZ?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 215: Bob-a-job

From New Scientist #1361, 9th June 1983 [link]

Arthur, Barney, Charlie and David are four small but intelligent cub-scouts. At the end of the last Bob-a-Job week each had raised over £1 (no halfpennies). Between them they had totalled exactly £5. Three had raised different amounts and Charlie’s figure was the same as Barney’s. These facts were known to all and each, of course, knew his own figure.

Was David’s figure, expressed in pennies, a perfect square? Charlie, had you got him to believe the wrong answer to that question, could have claimed to know on the strength of it that David had raised more than anyone else. But Arthur (whose total in pennies was not a perfect square) would not have believed the wrong answer, since he was already in a position to deduce the right one.

What precise sum had each one collected?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 1301: Magicless square

From New Scientist #2459, 7th August 2004

You are probably very familiar with magic squares such as:

in which you have to place the numbers 1 to 9 so that each row, each column and each of the two main diagonals add up to the same total.

Today your task is to do the opposite. Place the numbers 1 to 9 into the grid so that the three row totals, the three column totals, the two main diagonals and the total of the four corner entries are all different.

There are a few ways of doing this but, in order to retain some magic, do it so that the three-figure numbers formed by reading across the first row, by reading across some other row, and by reading down some column are perfect squares.

Find this magicless square.

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 214: Division. Figures all wrong

From New Scientist #1360, 2nd June 1983 [link]

In the following, obviously incorrect, division sum, the pattern is correct, but every single figure is wrong.

```          1 8 1
___________
3 2 ) 2 2 6 6 0
7 2      -----
3 4 6
2 8 9
-----
7 8
6 6
===```

Find the correct figures. (The correct division sum comes out exactly. The 3 figures in the answer are all only 1 out).

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 1302: Luncheon meet

From New Scientist #2461, 14th August 2004

As the crow flies, Joe lives 5 miles from each of his friends Ken and Les, who live 8 miles apart. Every Sunday lunchtime they meet at a convenient pub. It is convenient because the roads to it are straight and flat. They also chose this pub because its location meant they could all leave home at the same time and, after the shortest possible time, all arrive at the same time. Joe can only manage a steady 11 mph and Les a steady 14 mph on their bicycles. Ken, on his motor bike, is limited to 30 mph by the speed limit.

How long do they each take to reach the pub?

Note: I am waiting for a phone line to be connected at my new house, so I only have sporadic access to the internet at the moment.

## Enigma 213: Enigma’s square

From New Scientist #1359, 26th May 1983 [link]

SLA × SLA = ENIGMA
YSSE × YSSE = ENIGMAS

Usual letters-for-digits rules. The same letter is the same digit, different letters are different digits, throughout.

What (in letters) is the result of multiplying YE by EIYIELIGMI?

Note: I am in the process of moving house, so my time and internet connectivity will be constrained over the next couple of weeks. I’ll keep posting puzzles when I can. Normal service will be resumed when the internet comes to Wales.