Enigmatic Code

Programming Enigma Puzzles

Enigma 1231: A little list

From New Scientist #2387, 22nd March 2003

Here is a list of properties which a number may or may not have.

1. Several numbers divide into it exactly, but only one of them is prime.
2. It can be written as the sum of two or more consecutive positive integers.
3. It has more than two digits.
4. It is prime.
5. It is over 10000.
6. It is a perfect square.

You can choose any number and write down (in order) the numbers of the statements which are true. For example for the number 135 you would get statements 2 and 3 only, which written as one number gives 23.

There is just one number for which the list of true statements gives the number itself.

Which number?

Enigma 282: Beer bounty

From New Scientist #1429, 8th November 1984 [link]

Apart from providing the beer at my birthday party, I decided to encourage the abstemious by paying every guest a bounty of 10p for every pint drunk by someone else minus 20p for every pint he drank himself. I can’t remember all the details, but I do know that I paid out £9 in bounty altogether. Each guest drank a different odd number of pints, and everyone received some bounty money except David, who consumed so much beer (no one else’s consumption was within 3 pints of his) that he got exactly nothing in bounty.

From that information you can probably work out just how much each guest drank. If you do, please let me know.

Enigma 1232: Precocious segment

From New Scientist #2388, 29th March 2003

The digits on my calculator are made up of seven segments which light up in standard ways to form numbers.

For example, to form an 8 all seven segments light, but to form a 3 only five do. I have set the calculator to calculate to five figures after the decimal point, in which case it displays trailing zeros as well as a leading zero (for example it shows 1/25 at 0.04000) and rounds the last digit in the conventional manner. I have used it to calculate the reciprocals of prime numbers. All were correct as far as 1/13, but I immediately saw that the result for 1/17 was wrong although it looked like a valid number, and I realised that one of the 42 segments was lighting which should not.

If I continue to calculate the reciprocals of the next one thousand primes, how many more answers will be wrong?

Enigma 281: Family planning

From New Scientist #1428, 1st November 1984 [link]

Dick and Fecunda are family planning counsellors; so a bit of showing off is excusable. At any rate, the oldest of their five kids is called “Monday”, having been born on a Monday, the second is “Tuesday” (born on Tuesday), the third “Wednesday” (born on Wednesday), and so on. That makes the order of age clear enough but leaves the sexes more enigmatic.

Monday claims to be of different sex to Friday. Tuesday claims to have exactly three sisters; Friday claims to have exactly three brothers. Wednesday claims to have just one older sister and just one younger sister. Thursday claims to have a younger brother.

In fact there is at least one child of each sex. Any claims by one sex are true, whereas any by the other are false.

Can you list the sexes in the right order, starting with Monday?

Enigma 1233: Choosing the road

From New Scientist #2389, 5th April 2003

The road network in Philip’s country consists of roads running north-south at 1-kilometre intervals and roads running east-west at 1-kilometre intervals, dividing the country into squares. There are also roads along the two diagonals of each square. At each point where a north-south road meets an east-west road there is a tall signpost.

Philip uses the following method to get by road from one signpost to another. He stands at his starting post, points towards his finishing post and then chooses the road whose direction is closest to the way he is pointing. He goes along the chosen road to the next post and there he repeats the procedure, and so on to the finish.

Recently he made a journey which covered a distance of 300.5 kilometres, correct to the nearest metre.

To the nearest metre, how far apart were his starting post and finishing post?

Enigma 280: A division problem

From New Scientist #1427, 25th October 1984 [link]

Enigma 280

Find the missing digits.

Enigma 1234: Triangular grid

From New Scientist #2390, 12th April 2003

Triangular numbers are integers produced by the formula n(n+1)/2, like 1, 3, 6, 10. Replace the six letters in the triangular grid with digits (not necessarily all different) so that ABD, ACF and DEF are three different 3-digit triangular numbers and BC is a 2-digit triangular number. No number may start with a zero.

Enigma 1234

What is the numerical version of the triangular grid?

Enigma 279: Tall stories

From New Scientist #1426, 18th October 1984 [link]

I’ve just been taking to four interesting sisters. I asked the youngest how old she was. She whispered the answer to the tallest sister who whispered the answer to Anita who told me the answer was 53. I was a bit surprised and so I assumed that one of those three must have made a mistake. So I asked the eldest how old she was. She whispered the answer to Barbara who whispered the answer to the shortest who told me the answer was 26! Had these three made a mistake?

Charlotte then explained to me that although all four of them knew their facts and did their calculations correctly, before stating a number they always changed it. One of the sisters always doubled her numbers, one always added 1, one always reversed the digits, and one always squared her numbers and deleted the left-hand digit of that square.

The second eldest sister said she’d help. She asked the eldest and youngest their ages (and the eldest’s answer was the larger of the two) and she calculated the difference and told me what it was: I happened to know that this was also her age.

Then the youngest whispered her age to Davina who whispered it to the second tallest who told me.

What number did she tell me?

Enigma 1235: Power struggle

From New Scientist #2391, 19th April 2003

I have just given a class of students an interesting exercise. I wrote on the board the exact form of a particular number. Then I asked the class to calculate the sum of that number with its own reciprocal. They all got the correct whole number answer.

Then I asked the first student to square the original number and add that square to its own reciprocal. I asked the next student to cube the original number and add that cube to its own reciprocal. And so on around the class, increasing the power by one each time.

One of the students’ answers was 123.

What was the next student’s answer after that?

Enigma 278: Uncle Pinkle’s new system

From New Scientist #1425, 11th October 1984 [link]

Uncle Pinkle has moved house, and he is installing a new triphonic audio-system on his garden lawn. This system consists simply of three transmitters AB and C, each of which is to be exactly M yards from the others.

Uncle Pinkle will sit at a point X, carefully chosen so that the distances XAXB and XC are exact whole numbers of yards.

X must not be in a direct line with any pair of transmitters. [*]

The total XAXBXC is to be as small as possible.

Uncle Pinkle asks me to specify M, and the distances XAXB and XC, to meet his requirements.

Can you help please?

(If you find yourself using a calculator, let alone a computer, you are getting much too big. The rule in italics [*] is not as big a handicap as you may be thinking).

Enigma 1236: United share the title

From New Scientist #2392, 26th April 2003

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

Three points were awarded for a win and one for a draw. Albion had a clear points lead after the first Saturday, likewise Borough after the second Saturday, City after the third and Rangers after the fourth; and United were joint leaders after the final Saturday.

Knowing this you would be able to deduce with certainty the results of both the matches played on the final Saturday if I told you which matches were played on that day.

Which matches were played on the final Saturday and what was the result for each one?

Enigma 277: Professor Calendar’s sons

From New Scientist #1424, 4th October 1984 [link]

Professor Calendar (1930-82) was a noted mathematician. The almanacs published by him were well known. A census enumerator once asked him about the ages of his sons. The professor said: “All my three sons have the same date of birth and were born on the same day of the week”. He continued, “there are no twins and the date of today is the sum of their ages”. The enumerator, an able mathematician himself, replied: “Sir, I could not get their ages. Is one of them twice as old as his brother?”. The professor replied in the affirmative.

What was the age of the eldest son?

Enigma 1237: Odd socks

From New Scientist #2393, 3rd May 2003 [link]

George recently suffered a nightmare involving a seemingly bottomless drawer containing large numbers of socks of each of the seven colours of the rainbow. Old Nick whispered in George’s ear that if he selected two socks at random from the drawer the odds were exactly even that they would be the same colour.

The ghost of George’s mother-in-law then appeared, declared a distaste for violet socks, and threw out all the socks of that colour. With six colours remaining, the devil repeated that two socks selected at random had a 50:50 chance of being the same colour.

Four more ghosts appeared and threw out in turn all the indigo, blue, green and yellow socks. The Devil repeated his assertion each time, with reducing numbers of colours, that a pair selected at random had a 50:50 chance of being the same colour.

“There are 25 socks left, but how many have been thrown out?” George asked the Devil.

“I don’t give straight answers,” replied the Devil, “but I can tell you that it is an exact multiple of the original number of socks of your favourite colour.”

What is that multiple, and what is George’s favourite colour?

Enigma 276: Wrong figures to add

From New Scientist #1423, 27th September 1984 [link]

In the following addition sum all the digits are wrong. But the same wrong digit stands for the same correct digit wherever it appears, and the same correct digit is always represented by the same wrong digit.

Enigma 276

Find the correct addition sum.

Enigma 1238: Kindest cuts

From New Scientist #2394, 10th May 2003 [link]

I have a square carpet a whole number of feet wide. It is red and blue. Its design consists of a large blue square (not centrally placed, but with its sides parallel to the sides of the carpet) with the rest being red, making a small border around the square.

I need to trim the corners of the carpet slightly. To do this I shall make four straight cuts, one through each corner of the blue square. I shall discard the four triangular pieces leaving an irregular octagon. Given those constraints, I shall choose the angles of the cuts to leave me an octagon of the largest area possible. The effect of all this will be that I shall throw away a single-digit whole number of square yards of red carpet, being precisely 10 per cent of the red part.

How wide is the uncut square carpet, measured in feet?

Enigma 275: A few cross words

From New Scientist #1422, 20th September 1984 [link]

This was a clue in a recent mathematically inclined crossword:

“These two positive whole numbers add to 12 (two words, total 9 letters)”.

The answer could be FOUR EIGHT or NINE THREE (or, of course, EIGHT FOUR or THREE NINE).

In another crossword there was a similar clue; I’ll report it to you with a couple of omissions:

“These two positive whole numbers add up to ? (two words, total ? letters)”.

In this case the answer was unique (apart from the fact that the [order of the] two words could be reversed). The first of those omitted numbers is 12 or less, but even if you know what it was you would be unable to work out the second omitted number.

But now, knowing all that, if I told you the second omitted number you would be able to work out the first omitted number.

What are the two omitted numbers?

Enigma 1239: The next number…

From New Scientist #2395, 17th May 2003 [link]

Joe took a sheet of lined paper and asked me to write a whole number on the first line. This I did, writing a number that was less than a million and which consisted of the same digit repeated several times. After some elaborate calculations Joe wrote a number of the second line. After more calculations he wrote a number on the third line, and he continued like this until he had written a huge number on the eighth line. I asked him how he calculated each number. He explained, “If the number N is on one line then I work out the remainders when N is divided by 5 or 6. In some cases the first remainder was larger than the second and in those cases I wrote the square of N+1 on the next line. In the other cases the first remainder was smaller than the second and I wrote the cube of N+2 on the next line.” What number did I write on the first line?

Enigma 274: College angles

From New Scientist #1421, 13th September 1984 [link]

The heads of eight colleges are four married couples. The colleges are so sited that the chapel towers of all are visible from the towers of all the others. To ensure that the bedrock on which all are built is not shifting, each head daily at noon ascends his tower and checks the range and bearing of each of the others. It is interesting, is it not, that in doing so he or she never has to traverse his sextant by less than a quarter of a right-angle — nor of course by more than two right angles.

Each male head is a different distance from his wife. Mrs B is due east of Mr B, while Mrs A is due southwest of Mr A. Mr D is closer to Mrs D than Mr C to Mrs C.

Through what angle must Mr D turn his sextant in traversing from Mr C to Mrs C?

Enigma 1240: Stack ’em high

From New Scientist #2396, 24th May 2003 [link]

Joe’s daughter likes puzzles just as much as her father and spends quite some time designing her own.

Last weekend, while just fiddling with some standard dice, she worked out that she could stack them, one on one, so that the total obtained from the numbers on one face of the stack was the same for all four faces when she used the following rule:

Take the number on one face of the first dice, add to it twice the number on the corresponding face on the second dice, three times the number on the third dice, and so on.

She made a stack and showed it to her father. After some thought and a few calculations, Joe told her that by adding a few more dice his total would be six times hers.

How many extra dice did Joe have in mind?

Enigma 273: Moral choices

From New Scientist #1420, 6th September 1984 [link]

The campaign to Brace Up Moral Standards is run by a six-person committee, who elect a president from among themselves. Each person casts three votes. On the last occasion each voted for three persons and each person totalled a different number of votes.

Lord Luvaduck Love finished higher than Miss Marriage, who voted for herself. Tansy Tickell, the romantic novelist, fared better than the Reverend Cuthbert Custard, who did not vote for himself. Peregrine Prunes voted for Miss Marriage. Sylvia Slap, the agony aunt, voted for herself. Love did not vote for Tickell. No one votes for Love and Marriage; no one for Slap and Tickell; and no one for Prunes and Custard.

How many votes did each of these bracing persons receive?


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