Lightning cube roots

Umm, yes, sorry, got waylaid doing a new book. Here's what lies behind a trick I would perform for Year 9 (9th grade) kids. It was basically a stage act that is based on a very small amount of memorisation and a lot of trickery. Suppose I tell you I can supply the cube roots of every integer cube between 1 million and 8 million.

I give you a calculator, inviting you to enter a 3-digit number, less than 200, and read off the result after multiplying the number by itself twice. You might choose to enter the value 173, which would give you the value 5,177,717 for 173 x 173 x 173. When you read this out from the calculator, I would immediately tell you that the number you entered first was 173.

This does not involve memorising all the cubes up to 200³. That would also be possible, but it is unnecessary. You only need to memorise the cubes up to 20, because adding three zeroes will give you the cubes of all of the multiples of 10 up to 200. You can extend this further if you wish, but for this discussion, that is enough. My lower limit of 1 million means I can concentrate on 3-digit numbers, but again, the trick can be extended. I start with a table of cubes, where I only need an approximate value.

Here are the ten values I need to know:

number	100	110	120	130	140	150	160	170	180	190	200
cubed (millions)	1	1.3	1.7	2.1	2.7	3.3	4	4.9	5.8	6.8	8

 

There is a curious feature about the last digit of a perfect cube and how it relates to the cube root, the number we began with:

number ends in	0	1	2	3	4	5	6	7	8	9
cube ends in	0	1	8	7	4	5	6	3	2	9

 Now you have my secret: when you say “five million, one hundred…” I know we are between 170 and 180. Then I listen for the last digit of the cube (7), so I know the last number of the cube root is 3. QED, as Mr. Euclid might have said (if he spoke Latin): the answer is 173.

I thought I had invented this method, but I found out in 2019 that somebody called Wallace Lee beat me to it. Mathematics is like that!

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Sci: Keeping pill bugs

I apologise for the gap, but I  had two books to complete.

For several years, I had a small combination isopod-home-and-compost-heap sitting on my desk. This began as a test of an idea for my school students (if you came in part-way through this book, I was for some years a ‘visiting scientist’ in a local primary school). I wanted an easy way to set up individual pill-bug farms, and I worked with groups of three or four students to get their farms started. Let’s begin with my deluxe version, based on a clear polystyrene box that Ferrero Rocher chocolates come in.

A desktop compost heap which ran for two years.

For large-scale production in the school, we used thin plastic ‘takeaway food’ containers, and did the following, using sand from the school’s sandpit, and part of my home garden compost heap. Still, the clear chocolate box is better.

I take the box out to add extra leaves and sometimes a bit of water, but I often get the eggs or grubs of tiny flies called fungus gnats, and they can be a pest. Now you can work out why, most of the time, I leave the lid on when the box is inside.

A look inside: I used a dissecting needle to move the leaves aside to expose a resident. Inset: a clearer view

Instructions given to my students:

* Put about 6 to 10 mm of sand in the bottom and spread it out;

* Add just enough water to make the sand go dark (and note the colour difference);

* Get Peter to add some rotting litter (it may have germs, so he wears gloves);

* Using a brush and a tube, catch eight pill-bugs from the leaf litter from his garden;

* Add them to the container;

* Add some clean dead leaves for the pill bugs to eat;

* Put the lid on, and add a sticky label with your group’s names.

We also made five air holes in the lid with a needle. Those cheap containers split easily, but we drove all our holes through the label, which prevented splitting. After that, the students just needed to add water if the sand looked dry, and add leaves when the supply had dropped.

And that is how I invented the desktop compost heap. I have had one on my desk since then, and it is still doing well. At the time of writing, there is also a resident leech that has been parked there until I have time to photograph it. The pillbugs don’t seem to mind, and the leech emerges from the leaves to wave hungrily at me, each evening.

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Sci: A convection snake

Cut a piece of paper into a 6 cm diameter spiral. It doesn’t need to be too neat. I drew a guideline, and only followed it roughly.

Cut a piece of thread 15 cm long and tape one end of the piece of thread to the centre of the paper spiral.

Light the stove on low gas, and ask an adult to hold the paper spiral by the thread about 30 cm above the flame. (Caution: Do not allow the paper to catch fire.) What happens?

The energy from the gas flame heats the air above it. Warm air is less dense than cool air, so as the air heats up, it rises up from the stove. Cool air moves in to replace the warmer, lighter air. This “convection current” causes the spiral to twirl.

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Maths: primes and composites

Etching of an ancient seal identified as Eratosthenes.

The sieve of Eratosthenes

This is an ancient method (or algorithm) for finding prime numbers in a range of numbers. Write down all of the integers between 1 and n, then go through and cross out every second number starting at 4, every third number starting at 6, every fifth number from 10 on, seventh number beginning with 14 and so on. Some numbers will be crossed out more than once, but that doesn’t matter.

You end up with a sequence like this, where the bold numbers are the ones crossed out:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37…

Runs of composites

The aim here is to explore the natural sequence of integers for runs of consecutive integers that are composite, meaning they have factors. Two examples are series like 8, 9, 10, or 24, 25, 26, 27, 28. (Look at the example above: the crossed-out bold bunches are runs of composites.)

I started using a spreadsheet to help me search for runs longer than five composites, but for a while, the best I found was seven in a row: is there any link between the central number in the first example of any run of a particular size? Note that there will always BE a central number, as the totals will always be odd. Can you prove this? I can…

In October 2002, I found a run of 33 consecutive composite numbers, all of them less than 10,000. I used a spreadsheet to do it, and here are a couple of helpful hints.

* Note that I used the =IF function quite a bit…

* To test if a number x is exactly divisible by another number n, you use the form x/n=INT(x/n). I have some kludgy spreadsheet solutions, but no really good ones…

Play with it!

Because even numbers are always composites, and odd numbers are only sometimes composite, each string of composites will start and end with an even number. This means that every string will have an odd number of members. The numbers 8 to 10, 24 to 28 and 90 to 96 represent some of the strings that can be found with ease. For larger sequences, it is possible to construct a spreadsheet that will test for primeness, set flags for all composite numbers, and display longer sequences. Sadly, the margins of this book are too narrow for me to set the method out fully.

One way of generating a string of guaranteed composites is to take the series n! + (2, 3 … n). That is a tricky bit of notation. 5! = 5x4x3x2x1, and one string of composites will be 122, 123, 124, 125, but that is part of a larger string: that sequence is a guaranteed set, that's all. You do the rest.

Side note: n!-1 is often (but not always) prime, and the primes get rarer, once n exceeds 40.

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Maths: Pi from a spreadsheet

That is, π from a spreadsheet. This is heavy grade maths, under 15s should avoid it, unless they are very numerate.

There are quite a few series that converge on a value of π, or some function of π. Here’s one:

(π⁴)/96 = 1/1⁴ + 1/3⁴ + 1/5⁴ + 1/7⁴ + 1/9⁴ +…

These instructions will help you to create a spreadsheet that will get to π to about five decimal places in about 250 rows.

Begin with the value 1 in cell A2

Now from the home tab, select FILL SERIES (step 2) to get the numbers 3, 5, 7 … 199 in the cells down to A101. Column A is now ready.

Now enter this formula in B2 : =1/(A2*A2*A2*A2)

And put this in C2: =C1 + B2 (this will give us a running total of column B, up to that row).

Then put this in D2: =SQRT(SQRT(C2*96)) to get your first estimate of the value of π.

Now you can highlight cells B2, C2 and D2, and then highlight down to row 250, and use FILL DOWN from the Home tab to copy the formula down into those rows as well, and extend column A down to row 250 (think about this!).

Check the answers that you get in cells D246 and D247, after you have extended the spreadsheet down to row 250. The value you should be aiming at is 3.141 592 653 589 793 238 462… but getting there will take a while longer…J

If anything does not work, check the values in column A, which has to contain only the consecutive odd numbers. Then work your way across, checking each of the formulas in the instructions, until you spot the mistake you made. The formulas given here worked (many years ago) in MS Works and they still work in MS Excel (2010 version), but they have not been tested on other or more recent spreadsheets.

Here is what you will see: code on the left, output on the right.

Here, without explanation, are two other convergent series that close in on π.

π²/₆=1/1²+1/2²+1/3²+1/4²+1/5²+…

π/4=1-¹/₃+¹/₅-¹/₇+¹/₉…

There are more of these: use <infinite series pi> as your web search string.

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Maths: Amicable numbers

Charles Babbage’s Difference Engine, a forerunner of the modern computer. Lots of people have heard of Charles, but his son Herschel Babbage, a meticulous Australian explorer is hardly known.
Everything is connected!

The number 220 is the smaller number in the first pair of amicable numbers. The other number in the pair is 284. The proper divisors of 220 (1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110) sum to 284, while the proper divisors of 284 (1, 2, 4, 71 and 142) sum to 220. Up until 1946, there were 390 known pairs of amicable numbers, but by 2007, there were almost 12 million known pairs. The number reached 1,229,544,099, according the Amicable pairs list (https://sech.me/ap/ (last seen May 2025), which is available on the internet. It will be higher, by the time you read this…science is like that, and so is mathematics. 

Play with this.

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Arts/Sci: Seeing what you see

An ‘Artificial Spectrum Top’, devised by Mr. C. E. Benham, and sold by Messrs Newton and Co., furnishes an interesting phenomenon to students of physiological optics. The top consists of a disc, one half of which is black, while the other half has twelve concentric circles drawn upon it. Each arc subtends an angle of forty-five degrees. In the first quadrant there are three such concentric arcs, in the next three more, and so on; the only difference being that the arcs are parts of circles of which the radii increase in arithmetic progression. Each quadrant thus contains a group of arcs differing in length from those of the other quadrants. The curious point is that when this disc is revolved, the impression of different colours is produced upon the retina.

—Nature, 51 (1309), November 29, 1894, 113–114.

When toymaker Charles Benham invented his Benham discs, more than 130 years ago, he put these sorts of pattern on the upper surface of a toy top, which may give you a hint about how to spin it.

These small and simple devices will make you wonder just what colour really is. When you spin one, your eyes will often perceive colour. People say spinning the disc in the opposite direction can reverse some of the colours, and there are other interesting effects to find as well. I suspect that they have good imaginations, but why not test the effect out?

Two of Benham’s discs.

Use a pair of compasses and heavy paper to make discs like these, about 10 cm across. You can mount the disc on cardboard, fit a small bolt through the exact centre, and spin the disc in a drill (youngsters: get adult advice on using a variable drill).

The first report of the discs was a brief and anonymous note in the British science journal Nature in 1894. It described the disc as a black semi-circle, with a white half divided in four, and with black arcs on it. As the disc turns, it said, people see different colours from the different black arcs. Soon after Benham said that if you shine a bright sodium flame on the disc, you will see a very clear blue, and a very clear red, but other people said they could not see this at all.

The “official” explanation now says we have three kinds of light receptor in our eyes, in the same way there are three kinds of phosphor in a colour TV. Speaking crudely, these light receptors, the cone cells, are all sensitive to just one of red, green and blue.

According to the theory, you need all three kinds of cone in the retina of your eye to see colours normally. Somehow, the cones that pick up one of the colours (red, for example) must react differently to flashing lights of a particular frequency.

So with different size black bits on the disc, we get different frequency effects, and our eyes are stimulated to “see” different colours. That’s what the theory says, but nothing seems to explain the alleged effects of sodium light.

Some reports said different rotation speeds were needed for different people to see the same effects. Explore this claim, and see what you can discover. There are other patterns for Benham discs, some of them are better, some worse. Do some web research, then see if you can invent a better design.

Some illusions

Here are a few examples of the old standards that are all over the internet. The first two are from 19^th century German sources: a duck or a rabbit on the left and on the right, an old woman or a young one?

Duck or rabbit; and maid or crone?

Next: below, you will find an impossible cube and an impossible triangle: can you find their creators’ names?

Note:

Most of the time, two or three words will find your answer.  I used two searches: <cube illusion> and <triangle illusion>. (Note that when a search string is surrounded by angle brackets, you leave the brackets out.)

Where next?

To go further, you really should look into the works of M. C. Escher. Older readers can try Douglas Hofstadter’s Gödel Escher Bach: an Eternal Golden Braid which will teach you a lot about computers and systems. Now I warned you there would be some art in here: go and look up pointillism, and look at the work of Georges Seurat and Paul Signac.

Try pointillism yourself, although stippling, done with an Artline 0.3 mm fine-point is easier. My weevil (below) was done with a 0.3 mm Rotring pen with black ink, and there is not one line there.

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