Exercise of the Week #14

We’re back with some English after a little break!

Last Week’s Answers

This puzzle is known as a ‘cryptarithm’. Aside from being quite fun, they appear frequently in mathematical challenges, so it’s good to have a method for tackling them.

To solve cryptarithms, we must use logic and pattern recognition.

Note that F can only be 1, as even if both of the hundreds’ digits were 9, the maximum value for F would be 1.

Then, we know that the digits in the tens and ones’ positions must be the same.

We can use trial and error of values to work out the rest: given there are only a small number of possibilities, it shouldn’t take a prohibitively long time.

If we use these tips, we will find out that SUN + FUN = FANS represents 874 + 174 = 1048.

Exercise of the Week #11

English this week, returning with a longer-response comprehension question. Let’s make this a five-marker. Be as detailed as possible to avoid losing marks.

Last Week’s Answers

To solve this, we first need to decrypt each dubious doubloon stratagem!

1. Thomas Tew suggests they split the pile into twenty. So, each part is worth one-twentieth. Then he tells the others to take six parts: six twentieths. That would leave eight twentieths for Tew himself! There’s a difference of two twentieths (or one tenth) between what he’d be getting and what they’d be getting.

2. John Bowen suggests they each take half of 60% of the treasure. Half of 60% is 30%, and John would get the remainder - 40%! There’s a difference of 10% between his cut and theirs.

3. Bill Teach (that’s Blackbeard, by the way) says they should each take 0.75 of one half of the treasure. We need to convert the half into four-eighths: 0.75 (or three quarters) of this is three eighths. So, Teach would be getting four eighths: a one-eighth difference.

So, Tew would get one tenth more; Bowen would get 10% more, and Teach would get one eighth more. Who’s being most devious?

We need to convert these mixed measures so that we can compare them. Let’s make them all percentages.

We can start with one eighth: Take 100 and divide this by 8: to divide by eight quickly, just halve three times.

100 goes to 50 goes to 25 goes to 12.5. So, one eighth is the same as 12.5%.

Then, we can easily find one tenth as a percentage, as 1/10 is the same as 10/100, which is 10%.

And as 12.5% is bigger than Bowen and Tew’s 10%, Teach is the most devilish, and if his method was followed, he’d be awarded a whopping £12,500,000 more than the others!

Exercise of the Week #8

Back to English today; a model answer with guidance will be posted underneath next week’s exercise.

Last Week’s Answers

1)      I imagine you’ve written a cube! There are some much more complicated alternatives (such as a triangular bipyramid) but we don’t need to know about those for now…

2)      A cylinder is comprised of a rectangle and two circles. You can remember this as a cylinder is shaped like a tin: there’s a circular lid and base and a rectangular label.

3)      A pentagonal prism is effectively a 3D pentagon. The definition of a prism is a shape which has the same cross-section at all points along one of its dimensions. In this case, that shape is a pentagon.

4)      A square-based pyramid has five vertices and eight edges. Multiply these to get 40. A trickier one to spot.

5)      A sphere, when cut, will always give a circle. This circle is largest when you cut directly through the centre.

6)      Which 2D shape has angles which add to 720? You may know that this is a hexagon. Each of the internal angles of a hexagon is 120, and there are 6. As it looks like a rectangle from above, we must be thinking of a hexagonal prism.

Exercise of the Week #5

Today we’re looking at a comprehension question - the variety that’ll make a student shudder!

As ever, answers will be posted next week. Good luck.

Last Week’s Answers

You’ve probably seen this (or a similar problem) on your travels. It’s one of those where you might get a different answer each time you look at it, so is really an exercise is being systematic. This means employing a strategy to ensure you don’t miss anything.

So, let’s start with the smallest triangles.

As we can see, there are sixteen of those.

Don’t forget to include the triangles which are vertical - they still count.

Next up, we need to look at the medium-sized triangles: each one is comprised of four smaller triangles…

So now we’re up to 20. But wait, there are more of those medium-sized triangles! Don’t forget that they can overlap.

And now we’ve got 23. There aren’t any other medium-sized triangles, so now we’re onto the large triangles. These are made up of nine smaller triangles, and there are three of them.

So that’s it; surely we’ve got them all?

Not quite! There’s one more we mustn’t forget:

That’s right, the entire diagram is itself a triangle, which brings our final total to 27 triangles.

If you counted the little fellow on the top right as a triangle too, that’s fine! Technically he’s a square-based pyramid, or perhaps a triangle-based pyramid, so either way, that’s another four triangles there… eep.

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If you’d like a more in-depth explanation of this problem, as well as a complicated formula for working out the number of triangles in a similar diagram of any size, YouTuber ‘MindYourDecisions’ has an excellent video solution.

Exercise of the Week #2

Our exercise this week looks at persuasive writing. A model answer will be provided next week. Good luck!

Last Week’s Answers

1) She starts with £3000, but the King takes half, leaving her with £1500. However, he then gives her £500, which means she now has £2000. Then, the Chancellor takes one quarter, which is £500, leaving her with £1500.

2) Work backwards: if the Chancellor takes one quarter, Fred must have had £4000 after receiving money from the King. Now, take away the £500 he received. He’s got £3500. This is after the King has taken half - so we need to double it, giving us our final answer of £7000.

3) If I don’t have any money, the King takes half of 0, which is 0. But then, he gives us £500 regardless! But don’t get too excited: the Chancellor takes one quarter (£125) leaving us with £375. Nice!

4) If I’m left with £4125, this is after one quarter has been taken. So, £4125 is three-quarters of what we had before the Chancellor took his tithe. We need to know the full amount, four-quarters, so we can divide by 3 and multiply by 4 to give £5500. This is after the King gave us £500, so take that off: £5000. Finally, this is after the King halved our money, so double it: £10,000.