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Programming Enigma Puzzles
From New Scientist #1759, 9th March 1991 [link]
The Sunnycourt Tennis Club has 22 members, and each year it organises a Saturday afternoon tournament consisting of a number of singles matches. The matches are selected so that everyone in the club has at least one game, but the total number of games is always less than 22.
Last year, after the list of games had been settled, the secretary looked to see if there 8 games which involved 16 different players. As there were 8 courts available she had hoped to take a photograph with them all in use. Unfortunately, the list did not include 8 such games.
During this last winter, the club decided that this year’s tournament should involve one game fewer than that year. The secretary then did some calculations and found that however the games were selected, she was certain to be able to find 8 games in the list which involved 16 different players.
How many games were there in the tournament last year?
The Ruritanian prime minister is in a fix. Thanks to a series of incompetent policy decisions, all five of her senior ministers need to be axed from their posts. However, the PM cannot afford to sack them completely, because they will wreak havoc if moved to the back benches.
She has a solution: a reshuffle! She will move each of the five ministers to one of the other top posts, but no two of them will directly swap with each other. Anerdine will move to the department of the person who will become chancellor. Brinkman will replace the person who will be the new home secretary. Crass will take over the post being vacated by the person who will take Eejit’s job. Dyer will become health secretary even though he has been lobbying to become chancellor. The current defence secretary will get the department of the person who is becoming the education secretary.
Can you figure out who currently has which job, and where they are moving to?
From New Scientist #1758, 2nd March 1991 [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 out the sum with numbers substituted for letters.
From New Scientist #919, 17th October 1974 [link]
Hook, Line and Sinker returned separately from a day’s fishing and each reported his catch secretly (but accurately) to George Gudgeon, landlord of The Compleat Idiot. “Well, gents”, George announce later, “Hook caught most and Sinker least. Divide Hook’s by Sinker’s and you get Line’s”.
“Then I know how many each of the others caught”, Hook remarked on reflection.
“So do I”, said Line, after a pause.
“But I don’t”, Sinker complained, after a further pause.
“Never mind, old chap”, said George, “I’ll give you a clue. I’ve been out too and caught less than Hook. If you knew how many I caught, you could work out how many Line got”.
Sinker managed to work out Line’s total without a further word of help, this crowning a good day for him, in which he had caught twice as many fish as he had during the whole previous week.
How many did Hook, Line and Sinker each catch?
From New Scientist #1757, 23rd February 1991 [link]
Start with ** and write down its square. Then write down the next square and the next until the final one you write down is the square of **, each of the squares having three digits. Now count up all the occurrences of the digits in that list of squares; here’s an inventory:
Unfortunately the word processor seems unable to print digits, and replaces them with asterisks. But I can tell you that if the correct digits were here instead of the asterisks and you then took a grand count of the occurrences of the actual numerical digits in this Enigma (from “Start” to “list?” at the end) then, quite apart from the fours and fives and so on, there would be * noughts, * ones, * twos and * threes (where these last four digits are all different).
What was the two-figure number whose square started your list, and what was the two-figure number whose square ended your list?
Can you join the 36 dots below using 10 straight lines, with your pen never leaving the paper and no lines going outside the grid? At least one line must pass through each dot and no devious rule-bending is required. You will find lots of ways to do it with 11 lines, but 10 is much more of a challenge.
P.S. This puzzle forces you inside the box, but its solution is related to the classic four-line, nine-dot problem that was the origin of the cliché “thinking outside the box”.
From New Scientist #1756, 16th February 1991 [link]
As usual, letters have replaced digits in this addition sum, with different letters consistently being used for different digits:
We can also tell you that the damsel is in her prime, that is, SUE is a prime.
Find the value of the SUM.
From New Scientist #920, 24th October 1974 [link]
It was one of those gripping telly games where competing families have to pin famous names on famous faces. The faces were numbered 1, 2, 3, 4, 5, and the names, as listed by the Pettigrew family of Pinner, were Caesar, Beethoven, Newton, Shakespeare and Wedgewood Benn.
Also competing where the Quintons of Quilhampton, who proposed an order of Shakespeare, Wedgewood Benn, Caesar, Newton, Beethoven. The Ropes from Ruislip opted for Beethoven, Shakespeare, Newton, Wedgewood Benn, Caesar. The Surbiton Smiths listed Shakespeare, Beethoven, Caesar, Newton, Wedgewood Benn; and the Tophams of Tooting gave Shakespeare, Wedgewood Benn, Newton, Caesar, Beethoven.
As no two families got the same number right, the prizes presented no problem. Which of them went home to the envy of their neighbours with the first prize (a set of chrome and teak marrow-stuffers)?
From New Scientist #1755, 9th February 1991 [link]
There are a number of ways of moving about a grid. The diagram below shows one way of moving from the origin O to the point A.
The coordinates of O are (0, 0) and those of A are (4, 5). Like most normal grids, movement is permitted only along grid lines — diagonal moves are banned. But unlike normal grids, this one has its own particular rule governing movement — it is permitted only in positive directions, that is, to the right or upwards.
(a) the number of routes from O to A in the diagram above; and
(b) the number of routes, on an expanded grid, from O to a point B whose coordinates are (20, 6).
Catch Up 5 is a two-player game using five stacks of toy bricks of height 1, 2, 3, 4 and 5. The aim is to end with a taller tower than your opponent. Player A starts by taking a single stack of any height – in the example above, they chose the “2” stack. B then takes as many stacks as they want, piling them up until their tower is the same height or taller than A’s, which ends B’s turn. Here, B took the “1” stack, then the “5”. A now does the same, stacking until their tower is at least as tall as B’s. Here, A took the “3” stack, then the “4”. The players take turns until all the stacks of bricks have been used up, so A won this game.
Imagine you are going first in game against a Catch Up 5 expert who always plays the optimal move when it is their turn. Which piece should you choose?
From New Scientist #1754, 2nd February 1991 [link]
The 10 towns in Colouritania are joined by various roads. The map shows the 10 towns as small circles; I had drawn those roads I can remember. There are some other roads, each joining a pair of towns. There are no road junctions outside the towns, but one road can cross over another by means of a bridge.
Each day, for the past 1022 days, at 0600 hours, the 10 towns have each been allocated a colour, blue or yellow, with at least one town of each colour. The pattern of blue and yellow among the 10 towns has been different on each day.
On each of these 1022 days, at 0605 hours, the following descriptions have been applied: a road is said to be green if it joins a blue town and a yellow town; a town is said to be green if it has more green roads leaving it than non-green roads, or the same number of each.
Which of the following statements are true and which are false?
(a) We can say for certain there was a day when all the roads were green.
(b) We can say for certain there was not a day when all roads were green.
(c) We can say for certain there was a day when no road was green.
(d) We can say for certain there was a day when all the towns were green.
From New Scientist #921, 31st October 1974 [link]
Jumbo holidays announced a jolly fortnight by coach for married couples to Santa Quinina from the Costa Malaria. Despite the restriction, booking was so heavy that every room in every hotel in the place had to be reserved. This made the sums easy, since all the hotels were the same shape and size, each having as many floors as there were hotels and each taking as many guests on each floor as it had floors.
At the last minute some couples cancelled. After reshuffling, the top four floors of the Hotel Dolores were handed back with apologies and all possible spare coaches were laid off, otherwise everything went ahead as planned.
Each coach had seats for exactly four dozen holiday-makers and each holiday maker had a seat. (Drivers do not count, and there were no couriers).
How many empty seats were there on the coaches?
From New Scientist #1753, 26th January 1991 [link]
In a competition between three local football teams, the sides have played each other once. The points that they get for their matches are according to the new and, it is hoped, better system which the Lancashire football association has devised in the belief that it will lead to more goals. In fact, each side scored at least one goal in each match.
In this new system, 10 points are awarded for a win, 5 points for a draw, and 1 point for each goal scored.
I was originally told by a normally reliable source — the association’s secretary — that Anchester United had scored 11 points, that Boldon Wanderers had scored 14 points, and that Clackburn had scored 16 points. Unfortunately, however, the secretary later confided in me that due to his excessive consumption of local ale, each of the figure he had produced was incorrect.
Worse, all the secretary could remember was that these totals were in fact either 1 more or 1 less than the correct figure.
What was the score in each match?
The contestants were lined up, each hoping to get into the Millionaire chair. First, they would need to get through the “fastest fingers first” round.
The host cleared his throat: “List these animals in order of the number of legs they have, starting with the most:”
Guessing blindly, Jasmine went for CDBA, Virat chose CBDA and Finnbarr picked ADCB, but none got all four right. In fact, they all got the same number of answers in the correct position.
Which has more legs, a Fettlepod or a Sentonium?
From New Scientist #1752, 19th January 1991 [link]
In Susan Denham’s Enigma last week she related details of an unusual frame of snooker (and she reminded readers of the basic rules). Reading that teaser reminded me of a frame which I once saw between the two great players Dean and Verger.
Dean had a break; Verger followed this with a break of 1 point fewer; Dean followed this with a break of 1 point fewer; Verger followed this with a break of 1 point fewer; and so on, up to and including the final clearance.
In any one break no colour (apart from the red) was potted more than once. Dean never potted the blue. Each player potted the black twice as many times as his opponent potted the yellow. The brown was potted more times than the green.
How many times (in total) did the winner pot each of the colours? (Red, Yellow, Green, Brown, Blue, Pink, Black).
There are now 1443 Enigma puzzles on the site, along with 129 from the Tantalizer series, all 90 from the Puzzle series, and 87 from the Puzzle # series (and a few other puzzles that have caught my eye). There is a complete archive of Enigma puzzles published between January 1979 to January 1991, and from September 1997 up to the final Enigma puzzle in December 2013, which make up just over 80% of all the Enigma puzzles published. Of the remaining 345 puzzles I have 47 left to source (numbers 901 – 947).
In 2020, 100 Enigma puzzles were added to the site (and 43 Tantalizers, 5 Puzzles (which completes the Puzzle series), 53 Puzzle #s, and 2 others, so 203 puzzles in total).
Here is my selection of the puzzles that I found most interesting to solve over the year:
I have also been collecting old Teaser puzzles originally published in The Sunday Times on the S2T2 site, as well as accumulating my notes for more recent Teaser puzzles that I solved at the time. There are currently 409 puzzles available on the S2T2 site, 107 were added in 2020.
Here are some that I found interesting to solve (or revisit):
Between both sites I have posted 380 puzzles in total this year, bringing the total number of puzzles available to 2158.
Thanks to everyone who has contributed to the sites in 2020, either by adding their own solutions (programmatic or analytical), insights or questions, or by helping me source puzzles from back-issues of New Scientist.
As a bonus New Year puzzle you might like to try inserting mathematical symbols into the following countdown, to make the resulting expression equal to 2021:
10 9 8 7 6 5 4 3 2 1
Here are a couple of solutions:
(10 + 9) × 8 × (7 + 6) + 5 × (4 × 3 – 2 – 1) = 2021
10 × 9 + 8 × (7 × (6 × 5 + 4) + 3) + 2 + 1 = 2021
but there are many others.
From New Scientist #922, 7th November 1974 [link]
Little Arthur is full of curiosity. “Uncle Ebeneezer, please”, he enquired the other day, “Why is it that you sometimes say ‘O My Hat’, when you are vexed, and sometimes say ‘Flip’?”.
“I am not conscious of giving vent to either expression”, his uncle replied with dignity.
“I’ve heard you lots of times. But I can’t find any rhyme or reason. Why the difference?”
“Then I expect there is no difference”, replied the old codger, whose hobby is cryptarithmetic, “and I’m sure a few moments’ thought will show you that they come to the same thing”.
Can you detect the basis of this (correct) prediction?
From New Scientist #1751, 12th January 1991 [link]
In snooker there are 15 red balls worth one point each. If a player pots a red it stays in the hole and he (or she) is allowed to try to pot one of the colours yellow, green, brown, blue, pink or black (worth 2-7 points in that order). If a colour is potted it is brought out again and the player can try for another red, and so on. This continues until all the reds have gone. Then the remaining six colours are potted in ascending order.
The total points achieved in one such run is called a “break”. For example:
red + pink + red + black + red
would be a break of 16.
Having completed the break, the player sits down and lets the other player try for a red and continue the break, and so on. At the end the winning player is the one with the higher grand total of points. A player may choose not to pot the final black if the result is already determined without it. No other rules concern today’s story.
Stephens and Hendry play a frame of snooker. Stephens starts with a break of 3, Hendry follows with a break of 4, Stephens with a break of 5, and so on, and this pattern continues to the end. As usual in quality snooker, the black was potted more times that the yellow, and the pink was potted more times that the blue.
How many times did Stephens pot the brown? And how many times did Hendry pot the brown?
Lady Federica von Battenberg has baked a cake for her daughter Victoria’s birthday party. Eight children will be attending in all, so eight slices are needed.
She could, of course, make seven vertical cuts to make eight identical slices. But Victoria has heard it is possible to cut the cake into eight identical slices with only three straight cuts of the knife. In fact, there are at least two different ways to achieve this.
To be clear, not only must each slice be the same shape, they must all have the same amount of pink and yellow sponge and the same amount of marzipan on the outside.
Can you find two ways for Lady Federica to achieve this?
The major system is a centuries-old technique for memorising numbers in which they are converted into letters and words. Each digit is converted into a particular phonetic sound. For example “tch” represents one digit, and “n” another. Sometimes the same digit will represent different, but similar, sounds. For example “k” and “g” are represented by the same digit.
Each number below is shown alongside its conversion into the major system:
94146 → a partridge
84180 → four doves
07070 → six geese
125410 → ten lords
What numbers to the words “major” and “tune” represent, and why are they appropriate here? (It will all add up in the end).