Enigma 767: Safety in numbers

From New Scientist #1922, 23rd April 1994 [link]

Harry’s safe has a combination lock, with the usual dial with divisions marked round its edge from 0 to 99. Security rules forbid him to write down the combination, which he is expected to memorise. He has a poor memory for figures, so he can seldom remember it, but he can remember that it has four 2-digit numbers and one 1-digit number in sequence, and that if the sequence of numbers is strung together, the digits of the resulting 9-digit number are all different. He can also remember that if he subtracts his room number from that number, the result is a perfect square; of which the square root is a perfect cube; of which the cube root is his room number. He would not of course tell you all this, because he knows that it would enable you to work out the combination.

Reminding himself of his room number (which is on his door!) he can work out the combination in a few minutes, even without his pocket calculator (which reduces the time to mere seconds). He thus avoids the ignominy of having to plead with Security to open the safe for him.

You may take rather longer than Harry to work out the combination; but if I tell you that the sum of the digits of the cube is equal to its cube root, it could help cut the time.

What is the combination (the five separate numbers in sequence)?

[enigma767]

Enigma 774: Sting in the tail

From New Scientist #1929, 11th June 1994 [link]

In the above division sum sixes and zeros have been inserted wherever they appear. X stands for any digit, excepting 6 and zero but not necessarily the same digit throughout.

Find the value of the divisor XX.

[enigma774]

Enigma 773: Duodecimal

From New Scientist #1928, 4th June 1994 [link]

Two warnings about the sum shown here: the first is that it is a sum in base 12 throughout; the second is that in the bottom line the first O is the letter O (since no number starts with a zero), bit the subsequent O’s may be the letter O or the digit zero and the I may be the letter I or the digit one. Elsewhere O is the letter O wherever it appears and the I in FIVE is the letter I. Each letter stands for a different digit and the same letter represents the same digit wherever it appears.

If you want to have a chance of winning the tenner prize, please send in the value (in base 12) of FIVER.

[enigma773]

Enigma 772: Have you seen the trailer?

From New Scientist #1927, 28th May 1994 [link]

My friend was towing a small open trailer behind his car when the tow bar snapped. Luckily the trailer was light, so he turned it upside down and put it (like a hat) on the roof of his car. Consequently traffic behind could see his number plate twice, once on the car and once (upside-down) on the trailer. Amazingly, despite the inversion, both looked the same.

Being a British car, the number plate consisted of a letter, three digits and three letters (the first of the three letters not being I, O, S or Z for fear of confusing it with a digit).

The three digits were different and formed a three-figure number which was not a prime, not the sum of two primes, and not the difference of two primes.

What was the full number plate?

Note that by May 1994, the first letter (indicating the year of registration) had got as far as L, and the letter I was not used.

[enigma772]

Enigma 771: Cross the island

From New Scientist #1926, 21st May 1994 [link]

The map of the island shows the two ports, West and East, and the system of one-way roads — all traffic runs from west to east. The numbers show how many lorries are allowed along each road per day.

The island walking club wished to walk from the south [coast] to the north [coast] of the island. Each road has a toll bridge allowing walkers to cross it. The numbers on the map also indicate the toll for crossing each bridge. However the toll is only charged when a road is crossed from south to north — that is, when traffic is going from the walkers’ left to their right.

(a) What is the maximum number of lorries that can go from West to East in a day?
(b) What is the minimum cost the walkers have to pay if they choose their route carefully?

[enigma771]

Enigma 770: Long-distance calls

From New Scientist #1925, 14th May 1994 [link]

My telephone has a push-button dialling system in which the digits 1 to 9 are set out as shown above.

While using those buttons to telephone my friend Chris recently, I noticed that her six-digit telephone number had some interesting properties. It was an odd perfect square which used six different digits, and no two digits adjacent in her phone number were adjacent (horizontally, vertically or diagonally) on the telephone.

What is Chris’ phone number?

[enigma770]

Enigma 778: Trying triangle

From New Scientist #1933, 9th July 1994 [link]

Put a digit into each circle and read each side of the triangle clockwise, as a three-figure number. (For example, if I put a 2 in the top left-hand corner and continued, clockwise, 4, 6, 1, 4, 9, then the three-figure numbers would be 246, 614 and 492: notice that 492 is a multiple of 246, but 614 isn’t).

Your job is to choose the digits so that the second and third three-figure numbers are different multiples of the first three figure number.

What are the numbers in your triangle (clockwise from the top left-hand corner)?

[enigma778]

Enigma 777: An odd multiplication

From New Scientist #1932, 2nd July 1994 [link]

In the multiplication above, the digits have been replaced by letters and asterisks. Different letters stand for different digits but the same letter always stands for the same digit whenever it appears while an asterisk can be any digit.

[What is the numerical result of the multiplication?]

[enigma777]

Enigma 779: Cross-sums

From New Scientist #1934, 16th July 1994 [link]

The above square shows 16 digits forming two addition sums, 2091 + 1721 + 1406 = 5218 and 5112 + 2470 + 1029 = 8611. All the numbers have four digits, [and] no leading zeroes. In each addition sum the three different numbers to be totalled appear in descending order.

I have in front of me a similar cross-sum square, in which the numbers in each addition sum appear in ascending order. Not enough information? True. I need to tell you that the second numbers in the sums are larger than they might have been. What are they?

When this puzzle was originally published the numbers horizontally across the middle of the grid read “5 4 4 1” (not “5 4 2 1”), which does not give a working grid. Although this does not affect the solution to the puzzle.

[enigma779]

Enigma 776: Annual progression

From New Scientist #1931, 25th June 1994 [link]

Every year since I started teaching I have set the following arithmetic exercise to the class:

“Express this year’s number as the sum of two or more consecutive positive whole numbers”.

For example, 1970 = 491 + 492 + 493 + 494.

Each year someone in the class has managed to do it, but I suspect there will be a year in the future (possibly after I have retired) when, no matter how bright the class, no answer will be forthcoming.

What year will that be?

[enigma776]

Enigma 775: Length of time

From New Scientist #1930, 18th June 1994 [link]

Christine and Peter are going to clean the house. They can share the job or one of them can do it while the other goes shopping. If they share the job then the length of time they take is represented by the distance DE in the diagram. If Peter works alone then the length of time he takes is represented in the diagram by the distance AB.

Which distance in the diagram represents the length of time Christie takes if she works alone?

[enigma775]

Enigma 785: Brothers-in-law

From New Scientist #1940, 27th August 1994 [link]

Four married couples went together to the swimming pool and each person got a different locker number. Alan had the lowest with 846 and Janet had the highest with 875. Diane had 847. Christine had 865. Victor (who is not married to Diane) has 867, Steven had 869, Terence had 874 and I forget Nicola’s.

Among the eight people each person had at most one sibling in the group. The interesting fact was that, for any two people in the group, their locker numbers had a factor greater than one in common if and only if they were related as in-laws.

What is Nicola’s locker number?

[enigma785]

Enigma 784: The 3 circles game

From New Scientist #1939, 20th August 1994 [link]

The children have a new game. They draw three non-overlapping circles on the ground and put a number of stones in each one, say 10, 7 and 18. A move consists of:

(a) choosing any two circles, say C₁ and C₂, where C₁ contains fewer stones than C₂ or they both contain the same number of stones,

and:

(b) counting the number of stones in C₁ and then moving that number of stones from C₂ to C₁.

The children keep making moves until they empty one of the circles.

Here is a sample game, where, for each move, the chosen circles are underlined.

10, 7, 18
10, 14, 11
20, 14, 1
20, 13, 2
7, 26, 2
7, 24, 4
3, 24, 8
3, 16, 16
3, 32, 0

When the children begin a game they are never sure whether they are going to be able to empty a circle, particularly if the numbers are large.

Can you help the children by answering the following question: in which of the following games is it possible to empty a circle?

(a) 10, 7, 19
(b) 24, 16, 40
(c) 19, 13, 17
(d) 3741, 16255, 970843

[enigma784]

Enigma 783: A teeny enigma

From New Scientist #1938, 13th August 1994 [link]

Here is a crossword-type puzzle where you have to put a digit in each of the unshaded squares.

To help you I should give you clues for each of the three “across” numbers and each of the two “downs”. In fact the clue for each of them is “divisible by 18”.

Find the filled-in frame.

Unfortunately there is a misprint in the above puzzle. I had intended to fill in just one even digit for you and then there would have been a unique answer.

Find the filled-in frame which I expected.

[enigma783]

Enigma 782: Cubism on high

From New Scientist #1937, 6th August 1994 [link]

The Angel wore a puzzled frown: he had come across four-dimensional “hypercubes” when considering space and time; he knew that many people were familiar with real three-dimensional sugar cubes, but could not understand why a painter called Georges Braque, working in merely two dimensions, was called a Cubist.

Turning to his own research, he investigated fifth and higher powers. Soon he came across a number whose digits, raised to the same given power, then summed, gave the original number. He reported this to little effect. Turning back to his investigation, he discovered that the very next number had the same property. This proved a little more interesting, and when he pointed out the power concerned was not equal to the number of digits, he was instantly promoted to Archangel. After a long time, he despaired of finding another such example; and, still puzzled, has turned to studying Juan Gris and the Bergsonian concept of the dynamic universe?

What are the two numbers he found?

[enigma782]

Enigma 781: Structural count

From New Scientist #1936, 30th July 1994 [link]

In the picture the eight points A–H are joined together in one piece by seven lines between them. If I asked you to find the number of ways this could be done with A joined to 5 others, B joined to 2 others, C joined to 2 others, and the rest joined to one other then you should find that there are 30 different ways of doing it (including the one shown). I would not, for example, allow the seven lines to be:

AB, AC, AE, AF, AH, BC and DG

because then the picture would not strictly be joined up as it could fall into two parts with DG quite separate from the rest.

I now have in mind some particular positive integers a, b, c, d, e, f, g and h (with a the biggest down to h the smallest). If I asked you to find the number of ways of joining up A–H into one piece with seven lines so that A is joined to a others, B joined to b others, C joined to c others, and so on, then you should find that the number of ways equals the product (a×b×c×d×e×f×g×h).

What are the numbers a–h?

[enigma781]

Enigma 789: Fourth to first

From New Scientist #1944, 24th September 1994 [link]

Albion, Borough, City, Rangers and United have resumed their annual tournament in which each team played each of the other teams once. Two matches are taking place 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.

After three of the five Saturdays, the five teams have each gained a different number of points: Albion has gained most, followed in order by Borough, City, Rangers and United; but Rangers know that if they win their two remaining matches there are sure to end up with more points than any of the other teams.

(a) Which of the matches played on the first three Saturdays (if any) ended in a draw?

(b) Which match has been, or will be, played on the same day as Albion v. Rangers?

[enigma789]

Enigma 788: Count and count

From New Scientist #1943, 17th September 1994 [link]

Anna has a new counting game. She starts by writing a row of numbers, for example:

8, 3, 6, 11, 3, 4, 11, 3, 6, 3, 9, 9.

Then, thinking aloud, she writes down a description of the row as follows:

“The smallest number in the row is 3 and there are four 3s in the row, so I will write down 4, 3. The next smallest number in the row is 4, and there is one 4, so I will write down 1, 4.”

Anna carries on in this way until she reaches 11, the largest number in the row, and she sees there are two 11s in the row, and so she writes 2, 11. The complete row she has written down is:

4, 3, 1, 4, 2, 6, 1, 8, 2, 9, 2, 11.

Anna then repeats her counting with this new row. She gets:

2, 1, 3, 2, 1, 3, 2, 4, 1, 6, 1, 8, 1, 9, 1, 11.

She then repeats her counting with this new row, and so on.

Anna realises that once she has written down her starting row and got into her counting routine there are only two things that can happen. Either she eventually reaches a row that she has already had earlier and from then on she goes round and round a loop containing one or more rows, or she never gets a repeated row but continues for ever getting new rows.

Which of the following starting rows give Anna a repeated row?

(a) 57, 100.
(b) 10, 11, 12, 13.
(c) 1, 2, 3, 4, 10.
(d) 1000, 2000, 3000.
(e) 1000000.

There are now 1700 Enigma puzzles available on the site. Which leaves 92 Enigma puzzles remaining to post.

[enigma788]

Enigma 787: Red light

From New Scientist #1942, 10th September 1994 [link]

For my nephew’s birthday I gave him a novelty digital clock. The four-figure display is generally white but the clock can be programmed so that at any chosen times the display is red. When I gave it to him it was programmed to be red at the following times and white at all others:

(i) midday and midnight;
(ii) when the four-figure display uses precisely two different digits;
(iii) when the display uses four consecutive digits (in any order).

So, for example, the display was red at 12:43, it changed to white at 12:44, it stayed white until 12:59 and the it changed back to red at 01:00.

My nephew looked at one of the red times and read it as a ratio. He then programmed the clock so that the display was also red when the sum of the four digits displayed equalled his age (all of which times were previously white). The effect of this was that the ratio of the number of red times to the number of white times equalled the red ratio which he had read earlier.

How old is my nephew?

Happy New Year from Enigmatic Code!

[enigma787]

Enigma 786: Handshakes

From New Scientist #1941, 3rd September 1994 [link]

A number of people are standing in a large meeting room in a hotel. The oldest person in the room is 31 years older than the youngest person in the room, [and the age of the oldest person in the room was the same as the number of people in the room]. Each person shakes hands with every other person, but only once. If Adams shakes hands with Bolton, this is counted as one handshake.

After every person in the room has shaken hands with every other person in the room once, the total number of handshakes that has occurred is equal to the age of the youngest person in the room multiplied by the age of the oldest person in the room.

How many people are there in the room?

The following correction was published with Enigma 798:

The puzzle “Handshakes”, by Henry P. Dart III, as printed in the 3 September issue of New Scientist, lacks a key statement, namely, “That the age of the oldest person in the room [in years] was the same as the number of people in the room”. This then gives a unique solution to the puzzle that there were a total of [—] people in the room. The number of handshakes was [—]. The omission of the key sentence is regretted.

I have edited the correction to avoid given the solution, and the missing statement has been added to the puzzle text above (in square brackets).

[enigma786]