# Enigmatic Code

Programming Enigma Puzzles

## 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.

## Enigma 1303: Lucky numbers

From New Scientist #2462, 21st August 2004

Alex and his big sister Monica were using a new method to work out their lucky numbers. First they chose at random a number from 1 to 26 and assigned that value to the letter A. They then valued each of the remaining letters B to Z, with a number relating to its position in the alphabet. For example, if A had been given the value 10, then B would be 11, C would be 12, and so on through to Q=26, R=1 … Z=9.

With this exercise complete, they added up the totals for their respective names to find their lucky numbers. Alex was delighted, because (A+L+E+X) was bigger than (M+O+N+I+C+A). And they were both pleased to find that when the two lucky numbers were added together they made a perfect square.

What was the original value chosen for A?

## Enigma 212: More or less right

From New Scientist #1358, 19th May 1983 [link]

I displayed a three-figure number on my calculator. My son looked at the calculator upside-down and said:

“I can see a three-figure number too, and it’s less than yours”.

I added 12 to my number.

“I can still see a number, again less than yours”, he said.

I multiplied my latest number by 6.

“More this time”, he said.

“LESS”, he said, much to my surprise, and then we both laughed.

What number did I originally display?

## Enigma 1304: Some obvious facts

From New Scientist #2463, 28th August 2004

You know that:

• NINETY is divisible by 9.
• TEN is 1 more than a perfect square divisible by 9.
• There are SIX perfect squares between TEN and NINETY.

But in those displayed words each capital letter consistently represents a digit, with different letters used for different digits.

Which number should be SENT?

## Enigma 211: How’s that

From New Scientist #1357, 12th May 1983 [link]

The last ball was bowled; the stumps were drawn; and the epic cricket contest between All-Muggleton and Dingley Dell was over for another year. The twenty-two players determined to adjourn to a near-by tavern for refreshment. But they could by no means agree as to which of the four adjacent taverns was most deserving of their favour. At length Mr Pickwick proposed that each player should visit whichever two of the four establishments should best take his fancy; and so it was resolved.

In the ensuing hour just 11 players presented themselves at the Angler, just 11 at the Bull, just 11 at the Crow and just 11 at the Drum. The number who dropped in at both the Angler and the Bull was double the number who imbibed at both the Angler and the Crow and was not surpassed by the number of those who chose any other pair of taverns, which you can mention.

Afterwards it occurred to Mr Winkle to wonder how many players had availed themselves of both the Crow and the Drum. The company was by then too far advanced in high spirits to hit on the right answer and it was agreed to present the problem to you.

## Enigma 1305: Buzz

From New Scientist #2464, 4th September 2004

In the game of buzz the players form a circle and count in turn, the first saying “1”, the next “2”, the next “3” and so on. But every time the next number is a multiple of 7 or contains a 7 the player whose turn it is must say “buzz”, and then the direction in which play is going round the circle is reversed; so after the sixth player says “6”, the seventh says “buzz”, then the one who said 6 says “8” and the one who said 5 says “9” and so on until a player says “buzz” rather than 14, whereupon the one who said 13 must say “15”.

At 27 and 28, and again at 56 and 57, the direction of play is reversed twice in succession, and through the 70s two players must each say “buzz” five times alternately.

In a game which only ended when a player said “97” instead of “buzz” my only contribution was to say “buzz” twice.

(a) Which two numbers did I say “buzz” for?

(b) How many players took part?

## Enigma 210: Football: Letters for digits (4 teams)

From New Scientist #1356, 5th May 1983 [link]

Four football teams A, B, C and D played each other once. After some (or perhaps all) of the games had been played a table giving some details of the matches played, won and lost etc. was drawn up.

But unfortunately (Uncle Bungle again!) the digits have been replaced by letters. Each letter stands for the same digit (from 0 to 9) wherever it appears, and different letters stand for different digits. The table looked like this:

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

Find the score in each match.

## Enigma 1306: Three all

From New Scientist #2465, 11th September 2004

I have in mind three numbers each of three digits (no leading zero) in each of which one digit is 3. Of the following statements about them, three are true and three are false.

(a) The number is a prime.
(b) The number is (appropriately) a cube.
(c) The middle digit is the average of the other two digits.
(d) The third digit differs from the second by 3.
(e) The number has as a factor a two-digit prime the difference of whose digits is 3, or whose sum is a cube.
(f) The number belongs (appropriately) to the set of triangular numbers 1, 3, 6, 10, 15, 21…

What is the sum of the three numbers?

## Enigma 209: Double number slab

From New Scientist #1355, 28th April 1983 [link]

A double number slab is just two rows of squares in a rectangle, with the correct number in each square. The numbers in the top row are just 1, 2, 3, …, n. In each bottom row square is written the number of times the number above it occurs in the completed slab.

So, for instance, if n=5, you fill in the top row as shown, and in the bottom row you replace A by the number of 1’s, B by the number of 2’s, …, E by the number of 5’s in the whole slab.

Given that n is at least 4, can you say:

(a) for what value of n is it impossible to complete the slab properly?
(b) for what value of n is each second-row number less than or equal to every number to the left of it?

## Enigma 1307: KO, OK?

From New Scientist #2466, 18th September 2004

Sixteen players numbered 1 to 16 entered a men’s knockout tennis tournament. In each round the numbers of the remaining players were drawn at random to decide who played whom.

At the end of the tournament each player wrote down the number or numbers of the players he had competed against, in the order in which he had played them. The lists of the two finalists had their numbers in increasing order.

Also, each player worked out the total of the numbers in his list. The highest total was four times the lowest.

Which two players were in the final?