by Andrew Skidmore
Published Sunday March 08 2020 (link)
In our league three points are awarded for a win and one for a draw. Teams play each other twice per season. Comparison of the results of the two Divisions at the end of last season showed that
a. Division II held one more team than Division I.
b. The same total number of points was awarded in each division.
c. For the division with the larger number of draws, that number was equal to the number of matches not drawn in the other division.
d. The number of draws in one division was a multiple of the (3-digit) number of draws in the other.
How many teams were in Division II?
by Stephen Hogg
Published Sunday March 01 2020 (link)
Our local cemetery conservation lottery tickets have four numbers for a line. Using eight different numbers, my two lines have several symmetries. For each line: just one number is odd; there is one number from each of the ranges 1-9, 10-19, 20-29 and 30-39, in that order; the sum of the four numbers equals that sum for the other line; excluding 1 and the numbers themselves, the 1st and 2nd numbers share just one factor — as do the 2nd and 3rd (a different factor) and the 3rd and 4th (another different factor) and, finally, the 4th and 1st.
Printed one line directly above the other, my top line includes the largest of the eight numbers.
What is the bottom line?
by Victor Bryant
Published Sunday February 23 2020 (link)
I have some jigsaw-type pieces each consisting of one, two, three or four 1cm-by-1cm squares joined together without overlapping. The pieces are black on one side and white on the other, and they are all different. I have used all my pieces to simultaneously make some different-sized white squares in jigsaw fashion, with each square using more than one piece. Even if you knew what all my pieces were like, you would not be able to determine the sizes of my squares.
How many pieces do I have?
by Howard Williams
Published Sunday February 16 2020 (link)
While celebrating with friends in the pub, it was my round, and I worked out the number of ways that I could get everybody else’s drink wrong. I could remember the drinks (and they all wanted different drinks), but not who wanted which drink.
More friends arrived (there were fewer than twenty of us in total), and everyone still wanted different drinks. Again assuming that I could remember the drinks but not who wanted which drink, I worked out the number of ways that I could get everybody else’s drink wrong, and found that the number was 206 times the first number. Fortunately, it was no longer my round!
What was the second number?
by Bill Kinally
Published Sunday February 09 2020 (link)
Amelia noticed that 15 is equal to 1+2+3+4+5 or 4+5+6 or 7+8, so there are three possible ways that it can be expressed as the sum of consecutive whole numbers. She then told Ben that she had found a three-digit number which can be expressed as the sum of consecutive whole numbers in just two different ways. “That’s interesting”, said Ben. “I’ve done the same, but my number is one more than yours”.
What is Ben’s number?
by Peter Good
Published February 02 2020 (link)
A baker’s apprentice was given a 1kg bag of flour, scales and weights, each engraved with a different whole number of grams. He was told to separately weigh out portions of flour weighing 1g, 2g, 3g, and so on up to a certain weight, by combining weights on one side of the scales. He realised that he couldn’t do this if there were fewer weights, and the largest weight was the maximum possible for this number of weights, so he was surprised to find after one of these weighings that the whole bag had been weighed out. Upon investigation, he discovered that some of the weights weighed 1g more than their engraved weight. If I told you how many of the weights were too heavy, you would be able to work out what they all were.
What were the actual weights (in ascending order)?
by Stephen Hogg
Published January 26 2020 (link)
My “mystic” crystal stood on its triangular end face, with every side of the triangle being less than 10cm. It had three vertical sides and an identical triangular top end face, with the height being under 50cm. Unfortunately, during dusting, it fell and broke in two. The break was a perfect cleavage along an oblique plane that didn’t encroach on the end faces, so I then had two, pointed, pentahedral crystals.
Later, I found that the nine edge lengths, in cm, of one of these were all different whole numbers — the majority prime — and their total was also a prime number. Curiously, the total of the nine, whole-number, edge lengths, in cm, of the other piece was also a prime number.
What was the total for this latter piece?
by Danny Roth
Published January 19 2020 (link)
George and Martha and their five daughters with their families have moved into six houses in Super Street. I define a “super prime number” as a prime number which has at least two digits adding up to a prime number (e.g. 11 and 23). Similarly for “super squares” (e.g. 36) and “super cubes”. Houses in Super Street are numbered with the lowest 31 super numbers of the above types.
The elderly couple live in the highest-numbered house on the street. They noticed that the last digits of their daughters’ houses were five consecutive digits and the sum of their five house numbers was a perfect square. Furthermore, the ordinal positions (lowest-numbered house is 1 and so on) of all but one of the houses were prime.
Which five houses did the daughters occupy?
by Howard Williams
Published January 12 2020 (link)
Jenny is pleased that she has found two whole numbers with a remarkable property. One of them is a single digit greater than zero while the other one has two digits. The remarkable thing is that the difference of their squares is a perfect cube and the difference of their cubes is a perfect square.
Her sister Sarah is not impressed, however. She has found two three-digit numbers for which the difference of their squares is also a perfect cube and the difference of their cubes is also a perfect square.
In ascending order, what are the four numbers?
by Andrew Skidmore
Published January 5 2020 (link)
Liam had a bag of snooker balls containing 6 colours (not red) and up to 15 red balls. He drew out balls at random, the first being a red. Without replacing this he drew another ball; it was a colour. He replaced this and drew another ball. This was a red (not replaced), and he was able to follow this by drawing another colour. The probability of achieving a red/colour/red/colour sequence was one in a certain whole number.
After replacing all the balls, Liam was able to “pot” all the balls. This involved “potting” (ie, drawing) red/colour/red/colour…red/colour (always replacing the colours but not the reds), then “potting” the six colours (not replaced) in their correct sequence. Strangely, the probability of doing this was also one in a whole number.
What are the two whole numbers?