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?

]]>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?

]]>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?

]]>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)?

]]>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?

]]>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?

]]>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?

]]>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?

]]>Game show contestants are shown a row of boxes, each containing a different sum of money, increasing in regular amounts (eg, £1100, £1200, £1300, …), but they don’t know the smallest amount or the order. They open one box then take the money, or decline that and open another box (giving them the same choices, apart from opening the last box when they have to take the money).

Alf always opens boxes until he finds, if possible, a sum of money larger than the first amount. Bert’s strategy is similar, except he opens boxes until he finds, if possible, a sum of money larger than both of the first two amounts. Remarkably, they can both expect to win exactly the same amount on average.

How many boxes are there in the game?

]]>Beth and Sam were using computers to design two table mats shaped as regular polygons.

The interior angles of Beth’s polygon were measured in degrees; Sam’s were measured in grads. [A complete turn is equivalent to 360 degrees or 400 grads.]

On completion they discovered that all of the interior angles in the 2 polygons had the same numerical whole number value.

If I told you the last digit of that number, you should be able to work out the number of edges in (a) Beth’s table mat and (b) Sam’s table mat.

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