by Victor Bryant
Published September 15 2019 (link)
I have written down four different numbers. The third number is the highest common factor of the first two (ie it is the largest number that divides exactly into both of them). The fourth number is the lowest common multiple of the first two (ie it is the smallest number that both of them divide exactly into).
I can consistently replace digits by letters in my numbers so that the highest common factor is HCF and the lowest common multiple is LCM.
What are the first two numbers?
by Danny Roth
Published September 8 2019 (link)
George and Martha have a digital clock, which displays time with six digits on the 24-hour system, ie hh:mm:ss.
One afternoon, George looked at the clock and saw a six-digit display involving six different positive digits. He dozed off immediately, and when he awoke in the evening he saw another display of six digits, again all positive and different. He dozed off immediately and later on (before midnight) he awoke, having slept for exactly 23 minutes longer than the previous time. At that time, he saw a third display, yet again six different positive digits. He thus had seen eighteen digits and the nine positive digits had each appeared exactly twice.
At what time did George wake up after his first sleep?
by Bernardo Recaman
Published September 1 2019 (link)
The sum of the ages of six sisters known to me is 92. Though there is no single whole number greater than 1 that simultaneously divides the ages of any three of them, I did notice this morning, while they lined up for the ski lift, that the ages of any two of them standing one behind the other, had a common divisor greater than 1.
In increasing order, how old are the six sisters?
by Susan Denham
From Issue #1667, 3rd June 1989
The Anglo-Slovak club had its meeting last week. Those present were Tom, Vyctur, Ted, Tago, Ray, Min, Wex, Olav, Russ and Cy.
Some of the members stood up and took part in an old Slovakian dance, rather like a Morris dance. The dancers stood around the floor with no three in a straight line and between each pair a taut piece of ribbon was stretched across the ﬂoor. Some ribbons were pink and the rest were blue. I noticed that there was a pink ribbon between two of them precisely when their Christian names had an odd number of letters in common (So, for example, had a Jane, David and Victor been dancing, there would have been a pink ribbon from Jane to David, a blue from David to Victor, and a blue from Jane to Victor.) As soon as I saw how many dancers there were, I realised that two of the ribbons would have to cross. But they had arranged themselves in such a way that there was no pink and no blue triangle of ribbons.
Who was dancing?
by Ian MacKillop
From Issue #1666, dated May 27, 1989
I wrote to a mathematician friend in Utopia and asked him to send me the results of the recent general election there. He decided to make me work for it, as you can see from his reply:
- The Dextrous Party lost control of the Scitting (our 600-seat parliament) and now has fewer seats than the Sinistrals. The Other and Indeterminate parties remained third and fourth respectively;
- No new party was elected to the Scitting and none was removed;
- No party has an overall majority in the new Scitting;
- The Other Party lost almost half its seats, while the Indeterminate Party exactly doubled its seats;
- In the last Scitting all four parties held a perfect square of seats (the Other’s figure was also a perfect cube). In the new Scitting, two have perfect squares while the other two have perfect fifths (a whole number raised to the fifth power). No party had or has only one seat.
So now you can determine the composition of both old and new Scittings.”
by Andrew Skidmore
Published August 25 2019 (link)
I came across a most remarkable number recently and wrote it down. All its digits were different and it was divisible by 11. In itself, that wasn’t particularly interesting, but I wrote down the number of digits in the number and then wrote down the sum of the digits in the number.
I therefore had three numbers written down. What surprised me was that of the three numbers, one was a square and two were cubes.
What is the remarkable number?
by Susan Denham
From Issue #1669, 1st July 1989
One year, on my birthday I started a collection of thimbles. The following birthday l added to my collection, which went from strength to strength. In all subsequent years when I counted the thimbles on my birthday the total had increased from the previous year’s total by a number equal to the total I had on my birthday the year before that. (So, for example, my 1983 total equaled my 1982 total added to my 1981 total.) Now, by coincidence, my daughter was born on my birthday. And, with my collection growing following the described pattern, on our birthday in 1983 the number of thimbles I owned had reached exactly four times my daughter’s age on that day. On my birthday this year the total of thimbles was four times my age. On only one other occasion has the total been divisible by four, and that was in the year my son was born. How many thimbles were there in my collection on my birthday this year? How many (if any) did I have on the day my daughter was born?
by Keith Austin
Out there, somewhere in the night, is Elk Elloy, gunning for me. My only hope is to stay in the dark. Stretching ahead of me is the Boulevard, all all 3686.3 yards of it. If I can make the other end of it then I’ll be safe. But the whole length of the Boulevard is covered with neon strip lights, one hundred and ninety-three of them, each 19.1 yards long, set out end-to-end. They flash on and of steadily through the night. There go the 1st, 3rd, 5th, 7th, … 193rd. They are on for just an instant. Now there is a 12-second pause and then on come the 2nd, 4th, 6th, … 192nd for just an instant. Then another 12-second pause and we begin all over again with the odd numbered strips.
Fortunately, each strip only lights the ground directly below it, so there is a chance I can walk along the Boulevard and avoid ever being under a strip when it comes on. There are just two catches. First, I must walk at a constant speed which is a whole number of yards per minute, otherwise I will arouse the suspicion of Patrolman Nulty who covers the Boulevard. Secondly, I cannot walk at more than 170 yards per minute.
What speed should I walk at, in yards per minute?
by Stephen Hogg
Published August 18 2019 (link)
Using her ordinary 15cm ruler and Zak’s left-handed version (numbers 1 to 15 reading right to left) Kaz could display various fractions. For instance, putting 5 on one ruler above 1 on the other ruler, the following set of fractions would be displayed: 5/1, 4/2, 3/3, 2/4 and 1/5. Zak listed the fifteen sets starting from “1 above 1” up to “15 above 1”.
Kaz chose some fractions with values less than one from Zak’s sets (using just the numerals 1 to 9, each once only in her selection). Of these, two were in simplest form, one of which had consecutive numerator and denominator. Zak correctly totalled Kaz’s selection, giving the answer as a fraction in simplest form. Curiously, the answer’s numerator and denominator were both palindromic.
Give Zak’s answer.
by Eric Emmet
From Issue #1664, 13th May 1989
Four football teams are to play each other once. After some of the matches have been played a document giving some details of the matches played, won, lost and so on looked like this:
|Team||Played||Won||Lost||Drawn||Goals For||Goals Against||Points|
Two points are given for a win and one point to each side in a drawn match.
Find the score in each match.