# Enigmatic Code

Programming Enigma Puzzles

## Enigma 838: Time for insomniacs

When I went to bed last night I glanced at my digital bedside clock just before dozing off. When I stirred briefly between one and two hours later, I glanced at the clock again and confirmed that I had been asleep for that long. And then I had another sleep and when I stirred again, an exact number of hours after first dozing off, I again confirmed this on the clock.

When I can’t sleep I often ponder on the numbers displayed on the clock (like noting at bedtime one evening that the 1056 on the clock, as illustrated above, was the product of my age and my house number). On those three occasions when I looked at the clock last night, one of the numbers which I saw was palindromic, another was a perfect square and the other consisted of consecutive digits in increasing order.

In fact what I’ve just told you in the previous paragraph is rather misleading: it isn’t true about the three actual times because on the middle occasion I was very drowsy and I didn’t look at the clock but at its reflection in the bedside table top. I didn’t realise my mistake immediately because the number I saw did seem like a reasonable time.

At what time did I first doze off?

[enigma838]

## Enigma 937: Progressive football

From New Scientist #2092, 26th July 1997 [link]

The teams in the Midshires league play each other once during their season, which runs from mid-September to March. They each play up to three times a week, getting three points for a win and one point for a draw.

At the end of the past season, the teams’ total points were calculated and the teams were placed in decreasing order. The top team had one more point that the second which had one more point than the third, which had one more point than the fourth and so on until the penultimate team, which had one more point than the bottom team.

More than a third of the teams played in no draws at all, more than a third of the teams played in one draw, a quarter of the teams played in two draws, and the rest played in more than two.

How many teams are there in the league?

[enigma937]

## Enigma 837: The quick brown …

We arrange the 26 letters of the alphabet in a row as follows:

THEQUICKBROWNFXJMPDVLAZYGS (*)

Now take any letter, say P, and find the longest chains in (*) in alphabetical order ending with P. We find some of length 5, for example EIKMP, but none any longer. Next we find the longest chains in (*) in reverse alphabetical order starting with P. We find some of length 3, for example PLG, but none any longer. We say P has alphabetical length, α=5, and reverse alphabetical length, ω=3.

Question 1: I have chosen a letter which comes before P in the alphabet and to the left of P in (*). Can you say for certain my letter has α less than 5?

Question 2: I have arranged the 26 letters in a new row (**). Can you say for certain that if you choose a letter in (**) and I choose a different letter in (**) then they will have different α’s or different ω’s?

Now I want you to write on a piece of paper, a list of 25 possibilities for α and ω so [α=1 ω=1], [α=1 ω=2], …, [α=1 ω=5], [α=2 ω=1], …, [α=2 ω=5], [α=3 ω=1], …, [α=5 ω=5].

Next I want you to take each letter in (*) and work out α and ω for it and mark it on the list, for example you will write P against [α=5 ω=3]. Unfortunately, some letters will have a combination of α and ω that is not on the list, for instance X has α=6 and ω=4.

Question 3: Can you arrange the 26 letters of the alphabet in a row so that every letter has a combination of α and ω that is in the list? If your answer is “yes” then give such a row.

Question 4: Can you be certain that you will be able to find in my row (**) a chain of 6 letters that are in either alphabetical or reverse alphabetical order?

[enigma837]

## Enigma 836: Who buys the drinks?

A group of friends were in the Rose and Crown, debating who should pay for the round. By chance they were seated round the table in the sequence Alan, Brian, Charlie, David … with alphabetically consecutive first initials up to Mr Smith’s.

They took part of a pack of cards, shuffled it, and placed it face down on the table. They agreed to draw one card each, Alan, then Brian, and so on round and round the table, until someone drew a black card. That man would buy the drinks. If they had studied the cards first, they would have discovered that they had more than half the pack and that they had equal chances of drawing the first black.

In the event, the drawing process lasted as long as it possibly could with that selection of cards. What was the initial of the man who bought the drinks?

[enigma836]

## Enigma 835: Treble top

Whenever I play darts I keep track of my score by writing down how many points I scored for each go (consisting of three darts) followed by the number of that go. For example, if I scored 27, 154 and 84 on my first three visits I would have written 2711542843.

After a recent game I noticed that if the digits were consistently replaced by letters, with different letters for different digits, then it read:

TREBLETOPSATDARTS

My total score was a prime number.

[enigma835]

## Enigma 834: Square root of seven

This week I have replaced every letter of the alphabet by a digit. Of course, that means that some digits may represent several different letters, but any particular letter is replaced by the same digit throughout.

With my particular use of the digits you will find that each of the following seven:

TWO
THREE
FOUR
FIVE
SIX
SEVEN
EIGHT

is a number and that although one of the three-figure numbers is not a square all the remaining six numbers are perfect squares.

Find the value of ROOTS.

[enigma834]

## Enigma 833: Equal shares

Anna and Wesley each have seven book tokens and each token is for a whole number of pounds. The total value of Anna’s tokens is no more than £126 and the total of Wesley’s is no more than £127.

Anna sat down at the table and worked through all the possible combinations of her seven tokens, finding the total value for each combination. She did this by first taking each token by itself, then each pair of tokens, then each combination of three tokens and so on. She was looking for two combinations which had the same total. Wesley did the same thing with his tokens.

Anna had two cousins and she wanted to give each of them some tokens so that each received the same total value.

Wesley wanted to the same for his two cousins.

On the basis of the information we have, answer each of the following questions “Yes” or “No”:

(a) can we say for certain that Anna was able to find two combinations which had the same total?

(b) can we say for certain that Wesley was able to find two combinations which had the same total?

(c) Can we say for certain that Anna was able to give some tokens to her two cousins so that they each received the same total value?

(d) Can we say for certain that Wesley was able to give some tokens to his two cousins so that they each received the same total value?

[enigma833]

## Enigma 832: Big tax has little taxes

The country in which Uncle Fibo at present resides has introduced a peculiar new tax system. It is based on the following principles:

1. All taxable income is taxed at a “basic tax rate”.
2. On all taxes payed, a rebate is given, calculated at the basic tax rate.
3. All tax rebates are treated as taxable income.

Uncle Fibo has calculated that, at the basic tax rate applicable to him, the effective tax which he has to pay would be the same if he paid his entire taxable income for a given period over to the taxman and then received back from him a once-off non-taxable rebate on this amount, calculated at the basic tax rate.

What is Uncle Fibo’s basic tax rate, given to the nearest tenth of a percentage point?

[enigma832]

## Enigma 831: Pot black

In the snooker match between Davis and Whyte, the winner was the first player to win a certain number of frames, not greater than 16; the match ended as soon as one player had won the required number of frames. It is impossible for a frame to be halved.

Davies won all the frames played (including the first frame), whose numbers were perfect squares. Whyte won all the frames played whose numbers were primes; the sum of the numbers of the frames won by Davies was exactly the same as the sum of the numbers of the frames won by Whyte. Davies won the match.

How many frames did each player win?

Who won the penultimate frame?

[enigma831]

## Enigma 821: Diminishing returns

On my calculator each digit is formed by some of the seven liquid crystal strips being illuminated. For example the “8” uses all seven strips but the “1” only uses two.

Recently the calculator developed a fault. I displayed a single digit. Then at least one of the illuminated strips went out leaving a different digit displayed … Then at least one of the illuminated strips went out leaving a different digit displayed … I forget how many times this happened but eventually I read off the digits which I saw, and used them in that order to form one number. This number was not divisible by any of the individual digits used in it (in fact it was a prime).

What was the number?

[enigma821]

## Enigma 830: Post mix

In an inebriated state I tried to address two letters to each of five friends. I have listed my efforts below. Each envelope has four items:

– first name
– surname
– house name
– town

All the correct items occurred somewhere on the ten envelopes. But on each envelope the four items which I wrote all turned out to be from different people.

1: David Davis, Rose Cottage, Ely
2: Brian Clark, The Meadow, Carlisle
4: Alan Eyres, Waters’ Edge, Altrincham
5: Ed. Clark, Belle View, Carlisle
6: David Andrews, Waters’ Edge, Ely
7: Clive Brown, Belle View, Doncaster
8: Brian Eyres, Riverside, Doncaster
9: Clive Clark, Riverside, Ely
10: Ed. Davis, Rose Cottage, Carlisle.

What is Brian’s name and address?

[enigma830]

## Enigma 829: How do you manage without one?

My word processor has developed a fault. There’s one particular digit which it will not type: if you press the appropriate key absolutely nothing happens.

I can, for example, type the cube of 4 correctly, but not the square. Whereas for the larger number 78 (whose number of digits is less than the one-figure number I cannot type) I can correctly type both its cube and square.

Which digit do I keep missing out?

#### News

Between the Enigmatic Code and S2T2 sites there are now 3000 puzzles available.

On Enigmatic Code there are now 1658 Enigma puzzles available (which leaves 134 remaining to post). All 90 puzzles from the Puzzle series are available, as well as 215 from the Tantalizer series (and about 283 that are not yet posted). And we have all puzzles from the current Puzzle # series (which is ongoing, and most recently reached Puzzle #213).

And on the S2T2 site there are currently 840 Teaser puzzles available (these are also ongoing, and has just reached Teaser 3156, so there are quite a lot of those remaining. But I have been working through the published books of puzzles and newspaper archives that are available).

Along with a few additional puzzles that brings the total to the magical 3000.

If you have been playing along with me and have solved all the puzzles posted so far, then well done! It has been quite a journey.

As long as I have the time I will keep posting puzzles to the sites. Thanks to those who have contributed to the site, either by sourcing puzzles or sharing their solutions.

Happy Puzzling,

— Jim

[enigma829]

## Enigma 814: Mixed and matched

I had a day at the health club recently. I planned to have one full session of squash, one full session of badminton and one full session in the sauna (but not necessarily in that order) with at least a one-hour break between each of the sessions. The club’s timetable of sessions is shown above.

By coincidence my colleagues Mark and Jenny also spent the day there with the same idea in mind (although none of us necessarily did any of the activities together).

Our boss (who knew all the above facts) tried to telephone me at one stage but was told I was busy. From this he was able to work out my exact day’s schedule.

An hour later he tried to telephone Mark but was told he was busy and he was told the activity which Mark was engaged in. From this my boss was able to work out Mark’s exact schedule.

Another hour later he tried to telephone Jenny but was told she was busy and he was told the activity which Jenny was engaged in. From this my boss was able to work out Jenny’s exact schedule.

Which is the correct order of the men’s, women’s, and mixed sessions in the sauna?

[enigma814]

## Enigma 813: Easy as ABC

The schedule of matches has been drawn up for the next tournament between Albion, Borough, City, Rangers and United, in which each team will play each of the other teams once. Two matches will take place on each of five successive Saturdays, each of the five teams having one Saturday without a match.

Two of the five teams will be meeting their four opponents in alphabetical order. Given this information you could deduce the complete schedule of matches if I told you either one of the matches scheduled for the first Saturday.

1. Which teams will be meeting their opponents in alphabetical order?

2. Which matches are scheduled for the first Saturday?

[enigma813]

## Enigma 812: Upon my word!

I once had a job in a department store, working in the House Name Department. For example, I had to make “Dunromin” (taking 8 letters) and “Four hundred and twenty-one” (taking 23 letters).

On one occasion a customer ordered his three-figure house number spelt out in this way. I prepared the invoice and wrote on it (in figures) the number of letters used. But the invoice clerk thought this referred to the house number so he replaced it (in figures) with the number of letters that house number would take.

His superior again thought the number referred to the house number so he replaced it (in figures) with the number of letters that house number would take.

The auditor again thought the number referred to the house number so he replaced it (in figures) with the number of letters that house number would take. He then prepared the bill accordingly. The customer turned out to be very lucky: at each stage in this long process the number had been reduced and the bill was for less than half what the Es alone would have cost.

How many letters should the customer have been charged for?

[enigma812]

## Enigma 811: To change or not to …

Uncle Delroy had £3 for each of his nephews Sam and Tom and his niece Anna. For each child Uncle Delroy changed the money into 300 pence. He gave them 300 tests and they got a penny for each test they succeeded at.

Each test involved three boxes labelled 1, 2 and 3. While the child was not looking Uncle put a coin in one of the boxes. The child then chose one of the boxes. Uncle then chose one of the two other boxes; he always chose an empty box. He opened it and showed it was empty to the child. He then asked the child if he or she wished to change their choice to the other unopened box. After the child has either changed boxes or stayed with their original choice, they opened their final choice of box; if it contained the coin then they kept it otherwise the coin went to Uncle Delroy’s favourite charity.

Sam reckoned that Uncle showing him an empty box was no help so he never changed his choice. On the other hand Anna reckoned that uncle showing her an empty box did tell her something so she always changed her choice. Tom did not change in his first test but, for every other test, he changed if and only if he had lost in the previous test.

After the tests the following facts emerged. They each made a correct initial choice in the same number of tests. Tom lost in his first test, won in his last test and never won in two consecutive tests. Uncle paid out a total of £4.

How much did each child get?

[enigma811]

## Enigma 809: What’s the score?

From New Scientist #1964, 11th February 1995 [link]

In rugby union a try is worth 7 points if converted or 5 points if unconverted; a penalty goal or drop goal is worth 3 points. There are no other forms of scoring.

1. If in a match the winning side’s score is such that they cannot have scored any unconverted tries and the losing side’s score is such that they must have scored a converted try, what is each side’s score?

2. If in a match the winning side’s score is such that you know the number of tries that they scored and the losing side’s score is such that you cannot be sure of the number of tries that they scored, what is each side’s score?

3. If in a match the winning side’s score is such that they must have scored at least one penalty goal or drop goal and the losing side’s score is such that even if I tell you their score and the number of tries that they scored it is possible that you still cannot be sure how many of those tries were converted, what is each side’s score?

[enigma809]

## Enigma 810: Wooded acres

Having a few moments to spare one Thursday, I decided to measure the dining table. This is a fairly conventional piece of furniture in dark oak, rectangular in shape, the longer side less than double the width.

It emerged that whether the surface area is expressed in square yards, square decimetres, square feet, hectares or square light-years, the first significant digit is the same. Furthermore, whether the length of the table is expressed in yards, miles, millimetres or light-hours, the first significant digit is again the same one, and this applies also to the length of the diagonal.

The length of the perimeter is an exact whole number of half-inches, the area in square centimetres is an integral number which is a perfect cube, and the speed of light in my dining room is 0.3 kilometres per microsecond.

Please ascertain the width of the table in feet, to four significant figures.

[enigma810]

## Enigma 807: Track tickets

Susan is on a 2-day holiday, staying near one of the stations on the East-West railway line. There are stations every mile along the line. Tickets are sold in packs. The Go East pack contains a random selection of 9 tickets and each one is for travel of 1 or 2 or 3 or 4 or 5 miles in an easterly direction. The Go West pack contains a random selection of 5 tickets and each one is for travel of 1 or 2 or 3 or 4 or … or 9 miles in a westerly direction.

Susan’s plan for the first day of her holiday starts at her local station, where she will buy one pack of each kind. From the Go East pack she will choose a ticket at random and catch a train going east. When the ticket is used up she will alight; she will then take the pack for travel towards her home station, select a ticket at random, catch a suitable train and alight when the ticket is used up. She hopes to carry on in this way until she alights at a station she has been to before; she will then walk home.

The second day of her holiday again starts at her local station with her buying one pack of each kind. She will sort through the two packs and try to find a selection of East tickets and a selection of West tickets so that the total distances for the two selections are the same; she will then use all the selected tickets to make a trip that ends at her home station.

Q1. On day one, is there a possibility Susan might find herself on a station, wanting a ticket in a particular direction and having none of those left?

Q2. On day one, is she certain eventually to reach a station she has been to before?

Q3. On day two, is she certain to be able to find selections of tickets that meet her requirements?

[enigma807]

## Enigma 808: Multi-choice quest

In a recent maths test five statements were labelled A to E. Statement A was “it is an odd two-figure number”, and statement C was “it is a two-figure number which is an odd perfect square”.

The other three statements (whose order I forget) were:

“it is a two-figure prime”,

“it is a two-figure number which is the sum of two consecutive numbers”, and

“it is a two-figure number with one odd digit and one even”.

Candidates who chose an easy question were given a particular two-figure number and asked to tick which applied:

A is true
B is true
C is true
D is true
E is true

For those whose chose a hard question the boxes had the following different headings and the candidates were asked to tick the box or boxes which were always true:

B implies C
C implies D
D implies E
E implies A
A implies B

The box or boxes which should be ticked were the same for both questions.

What was statement D? And which of statements A to E were true in the easy question?

There are now 1650 Enigma puzzles available on the site, and 142 remaining to post. So there is currently around 92% coverage.

[enigma808]