Can a 747 take off from a conveyor belt runway?

This question is currently going round on Facebook; one of my friends posted it, I answered it, and a few hours later entered a lengthy and circular debate in the comments section.  Here, with more space than a Facebook comments section, I'd like to pose the question, answer it and then address some of the misconceptions.

Here's the question:  Imagine a 747 is sitting on a conveyor belt, as wide and long as a runway.  The conveyor belt is design to exactly match the speed of the wheels, moving in the opposite drection.  Can the plane take off?

Shared originally by Aviwxchasers.Com a news and media site from the US.

My friend shared the post, and answered, "No - there's no way to generate lift."
My original answer:  

"I think you're assuming that the 747 will be moving because its wheels are being driven. The forward thrust of the aircraft comes from its engines, not from making its wheels turn (like a car). So the engines push the aircraft forward and the wheels just slide and skid along the conveyor belt , while the engines push the aircraft to take-off speed. The wheels don't have to turn to allow the aircraft to move."

However, this wasn't deemed sufficient by some other commentators, who (to summarise) posted the following questions or objections.

1. The conveyer would counteract the forward movement produced by the engines' thrust.  The conveyor matches the speed of the wheels in the opposite direction, so there can be no forward motion.
2. Lift is created by air flow over the wings. It doesn't matter how much thrust you have , if doesn't generate airflow over the wing, it won't fly.
3. The method of propulsion should be irrelevant. If the speed of the conveyor matches the speed of the wheels in the opposite direction, there can be no forward motion.
4. Take the vehicle out of the equation, it's all about the wheels and the conveyor. The faster the wheels turn, the faster the conveyor goes. Stick anything you want on top of the wheels, the principles are the same. Newton's third law
5. Indeed it's not about the wheels it's about forward movement (thrust from the engine) needed to take off but that forward movement is being counteracted by the conveyer    

Here, I propose to look at these arguments individually and collectively, and explain why the points which are raised aren't sufficient to stop the aircraft from moving and taking off.

1.  The conveyor would counteract the forward movement produced by the engines' thrust.  The conveyoy matches the speed of the wheels in the opposite direction, so there can be no forward motion.

1A.  The conveyor will only stop the wheels' rotation from generating forward movement.  However, it is not the wheels which are being driven - this is not a car or a truck.  The forward movement of the plane is not a result of the wheels successfully pushing against the conveyor belt; the forward movement of the plane comes from the push of the engines.  The result is that the plane will move forwards (the engine pushes against the air flowing through it, the air pushes back - Newton's Third Law) and the wheels will spin and skid down the track.

It's not "The plane moves forwards because the wheels go round", it's, "The wheels go round because the plane goes forwards [over a surface which has sufficient friction] ."


2. Lift is created by air flow over the wings. It doesn't matter how much thrust you have , if doesn't generate airflow over the wing, it won't fly.

True.  But there is thrust, and it is generating airflow over the wing.  The conveyor belt does not have the ability to resist the movement of the plane, only to counteract the turning of the wheels.  

The plane can move forwards -even along the ground - without its wheels turning.  The engine isn't driving the wheels.

3.  The method of propulsion should be irrelevant. If the speed of the conveyor matches the speed of the wheels in the opposite direction, there can be no forward motion.
and 
4. Take the vehicle out of the equation, it's all about the wheels and the conveyor. The faster the wheels turn, the faster the conveyor goes. Stick anything you want on top of the wheels, the principles are the same. Newton's third law

3A and 4A.  Comment 3 was a response to my question, "What happens if we replace the 747 with a space rocket, aligned horizontally on the conveyor belt runway, on wheels?"  and the comment suggests a misunderstanding about what's causing the forward motion of the plane.  The thrust of a space rocket is completely independent of the any wheels (they normally fly fine without them) and the wheels will just get dragged along the conveyor belt, without turning. 

Yes, the faster the wheels turn, the faster the conveyor belt goes.  But the wheels don't have to turn for the plane to move (as I said in response to 2).  The conveyor belt does not have some property which prevents something from skidding along it, just from preventing any forward motion due to wheels turning on it. 

You can only take the vehicle out of the equation when you realise that the vehicle isn't a car, with the limitations that a car has.

Newton's third law applies to rotating wheels and conveyor belts.  It also applies to the aircraft engines and the air flowing through them, or to space rockets and the fuel burning inside them.  I wonder if the Starship Enterprise (on wheels) would have this problem?

5.  It's all about forward movement (thrust from the engine) needed to take off but that forward movement is being counteracted by the conveyer  

The forward movement is not counteracted by the conveyor.  The conveyor just stops rotating wheels from generating any forward movement.  But if the wheels aren't rotating (because they're not being driven, such as in a car) then there's no movement to resist.

In conclusion, I'd like to offer this video from "Mythbusters" which shows a light aircraft attempting to take off against a conveyor belt (which in this case is being pulled by a pickup truck to match the plane's speed).  To quote one of the engineers, "People just can't wrap their heads around the fact that the plane's engine drives the propellor, not the wheels."

Some other 'everyday maths' articles I've written:

A spreadsheet solution - the nearest point to the Red Arrows' flight path from my house
The Twelve Days of Christmas - summing triangle and square numbers
Why are manhole lids usually circular?
"BODMAS" puzzles - what's the fuss?

Numbers from 1,2,3,4 to 50

As an occasional diversion from work, I like to try maths puzzles, and to this end, I recently purchased a couple of high school maths textbooks (for use with students aged from about 14 to 16).  Not that I can't do the puzzles (I think my maths is okay), but because sometimes the questions and puzzles in them some interesting ideas about extension activities - for example, the recent question about the circle in the corner of a circle and a square is probably intended to be solved with trigonometry, and instead, I solved it in terms of square roots, using just Pythagoras' theorem.  That led to me solving the situation for the circle in the corner of a hexagon (which wasn't in the textbook, but which had an interesting solution too).

Anyway, I found a great question in the extension section at the end of one of the textbooks, and it goes like this:

Creating Numbers: a task requiring imagination

Your task is to create every number from 1 to 50.
You can use only the digits 1, 2, 3 and 4 once in each and the operations + - * / . 
You can use the digits as powers, and you must use all of the digits 1, 2, 3, 4

Here are some examples:

1 = (4-3) / (2-1)
20 = 4² +3 +1
68 = 34 * 2 * 1
75 = (4+1)² * 3


So, here goes... 1 to 50, using only 1, 2, 3 and 4 and the basic maths operators.  Some of the answers seem a little repetitive or derivative (look at 38 through 43), and in some cases I found alternative answers afterwards.


1 = (4-3) * (2-1)
2 = (4-3) + (2-1)

3 = (4+2) / (3-1)  
4 = (4 * 3) / (2+1)  
5 = (2 * 4) – (1 * 3)  
6 = (2 * 4) – (3-1)
7 = (3+4) * (2-1)
8 = (3+4) + (2-1)  
9 = (3+4) + (2 * 1)  
10 = 1 + 2 + 3 + 4  
11 = (4 * 3) – (2 - 1)  
12 = (4 * 3) * (2-1)  
13 = (4 * 3) + (2-1)  
14 = (4 * 3) + (2 * 1)  
15 = (4 * 3) + 2 + 1
16= 4 ^ ((3+1)/2)  
17 = 3(4+1) + 2  
18 = 4² + (3-1)  
19 = 4² + (3 * 1)  
20 = 4² + 3 + 1  
21 = (4+3) * (2+1)  
22 = (4+1)² – 3  
23 = 3² + 14  
24 = 1 * 2 * 3 * 4  
25 = 31 – (2+4)  
26 = 13 * (4-2)  
27 = 3² * (4-1)  
28 = 32 – (4 * 1)  
29 = 31 – (4 -2)  
30 = (4+1) * 3 * 2  
31 = 34 – (1+2)  
32 = 4^(3-1) * 2  
33 = 34 – (2 -1)
34 = 34 * (2-1)  
35 = 34 + (2-1)  
36 = (4 * 3) * (2+1)  
37 = 34 + 1 + 2  
38 = 42 – (3+1)  
39 = 42 – (3 * 1)
40 = 41 – (3-2)  
41 = 43 – (1 * 2)  
42 = 43 – (2-1)  
43 = 41 + (3-1)  
44 = (14 * 3) + 2  
45 = 43 + (2 * 1)  
46 = 42 + 1 + 3  
47 = 41 + (3 * 2)  
48 = 24 * (3-1)  
49 = ((4 * 1) + 3)²  
50 = 41 + 3²

As the logical extension, I attempted to carry on past 50.  It becomes increasingly difficult, since 1, 2, 3 and 4 are all small numbers, and the combinations of those small digits become less useful in making specific larger values (especially the prime numbers). 

However, if we expand the rules to allow ! (factorial) and decimal points, then this enables us to find solutions for 57 (for example).  I'd like to thank the free math help forum community (especially Denis for his initial suggestion to extend the rules), for the additional solutions, comments, corrections and suggestions.  They've been very friendly in quickly adopting my idea and sharing their comments and solutions.  An additional rule that's been introduced is the use of decimals - by doing this, we can include dividing by .2 (for example) as a way of multiplying by 5.

51 = (12 * 4) + 3  OR (4²+1) x 3
52 = 4³ - 12
53 = (1 + 4!) * 2 + 3
54 = (13 * 4) +2
55 = 34+ 21
56 = (1 + 3 + 4!)2
57 =(1+4)! / 2 - 3  OR (4 + 2) / .1 - 3
58 = (31 * 2) - 4
59 = (21 * 3) - 4
60 = 3⁴ - 21
61 = 4³ - (1 + 2)
62 = 4³ - (1 * 2)
63 = 4³ - (2 - 1)
64 = (2 - 1) * 4³
65 = (2 - 1) + 4³
66 = (2 * 1) +4³
67 = (34 * 2) -1
68 = 34 * 2 * 1
69 = (34 * 2) + 1
70 = 4³ + (1 + 2)!
71 = ((4! / 2) * 3!) - 1
72 = 24 * 3 * 1
73 = (3 * 4!) + (2 - 1)
74 = (3 * 4!) + (2 * 1)
75 = (4+1)² * 3
76 = (41 * 2) - 3!
77 = ((4! + 1) * 3) + 2
78 = (4! + 2) * 3 * 1
79 = 3⁴ - (2 * 1)
80 = 3⁴ - (2 -1)
81 = (4! + 1 + 2) * 3
82 = 3⁴ + (2 - 1)
83 = 3⁴ + (2 * 1)
84 = 3⁴ + 2 + 1
85 = (43 * 2) - 1
86 = 43 * 2 * 1
87 = (21 * 4) + 3
88 = (43 + 1) * 2
89 = 1+ 2^(3!) + 4! 
90 = (1! + 2!) * (3! + 4!)
91 = (23 * 4) - 1
92 = 23 * 4 * 1
93 = (23 * 4) + 1
94 = (1 + 3)4! - 2
95 = 3!*(2^4) - 1
96 = (12 * 4!) /3
97 = 4(3! - 2)! + 1
98 = (1 + 3)4! + 2
99 = 123 - 4!
100 = (3 / .12) * 4


The original textbook question asked only for the numbers from 1 to 50 with only powers and basic operators, but by expanding the rules, many more numbers have become achievable.  Dare we go above 100?

Further posts:  Numbers 101-150 and Numbers 150-200.

If you enjoyed this puzzle, you may also be interested in some of the other puzzles I've explored more recently:

Snakes and Ladders (the Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams with five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero in as few steps as you can)
Calculator Games: Front to Back
Calculator Games: Up, up and away with Ulam sequences
Calcualtor Games: The Kaprekar Constant

What is a "growth hacker"?

Okay, I admit it: I'm confused.  I kept up to date with "experts", "gurus", "rock stars" and "ninjas", but I've reached the limit of my understanding. Why are we (as the online analytics community) now using the term 'hacker'?  When modems were dial-up, and going online meant connecting your computer to your telephone headset, hackers were bad people who illegally broke into (or 'hacked') networks

Nowadays, though, hackers are everywhere, and one of the main culprits (especially online) are the "growth hackers".  I'm just going to borrow from Wikipedia to set some context for what these new hackers actually are:

• Hacker (term), is a term used in computing that can describe several types of persons
  • Hacker (computer security) someone who seeks and exploits weaknesses in a computer system or computer network
  • Hacker (hobbyist), who makes innovative customizations or combinations of retail electronic and computer equipment
  • Hacker (programmer subculture), who combines excellence, playfulness, cleverness and exploration in performed activities

Hacker image credit: Pinsoft Studios
So we have, "excellence, playfulness, cleverness and exploration" in performed activities?  Really?  That's what a hacker is nowadays?  Can't we just be good at what we do?  We have to be playful and creative while we're at it?  Perhaps I'm an optimisation hacker and I never realised.

My research into growth hacking indicates that the term was first coined in 2010 by Sean Ellis.  Why he chose 'hacking', I'm not sure (especially given its previous connotations), but here's how he described growth hacking:

"A growth hacker is a person whose true north is growth.  Everything they do is scrutinized by its potential impact on scalable growth. ... I’ve met great growth hackers with engineering backgrounds and others with sales backgrounds; the common characteristic seems to be an ability to take responsibility for growth and an entrepreneurial drive.  The right growth hacker will have a burning desire to connect your target market with your must have solution ...  The problem is that not all people are cut out to be growth hackers."

So:  a growth hacker is a marketer whose key performance indicator is growth.  So why 'hacker'?  Perhaps it's about cracking the code for growth and finding a short cut to success?  Perhaps it's about carving a way through a jungle filled with bad ideas for growth, with an instinctive true north and a sharp blade to cut through all the erroneous ideas?

Image credit: Recode.net

It's an imaginitive either way.  And now, five years after Mr Ellis's post, it seems we have growth hackers everywhere (ironic, considering the URL for the original blog post is /where-are-all-the-growth-hackers/  - they now have multiple websites and Twitter accounts ;-)

So - my curiousity has been satisfied: a [good] growth hacker is a marketer who will help rapidly accelerate growth for a small or start-up company by rapidly analysing what's working for its audience and focusing on those strategies with agility and velocity.  Why are they so popular now?  Because - I'm guessing - after the global financial issues of 2008-2009, there is now much more interest and emphasis in start-ups and the entrepreneurial spirit - and every start-up needs a growth hacker to crack the code to accelerated growth rates.

A circle in the corner of a hexagon

In my last post, I calculated the radius and diameter of a circle drawn within the external 'corner' of another circle - diagram below.  The final ratio indicated that the diameter of the small circle is 10.8% of the larger circle.  In this post, I'm going to extend this to look at circles in hexagons (something that I discussed a few years ago when I looked at square or cubic packing and hexagonal packing).  I'll also then look to apply my findings to real life, by reviewing the sizes of atoms in alloys - how do they fit together?

Firstly, here's a reminder of the result from last time:

For a circle with radius DE = EF = 1 unit, then the radius of the smaller circle "in the corner" BC = CD = 0.108... units.  The smaller circle has a radius of 11% of the larger one.  The length AD is 40% of the radius of the larger circle.

In this blog, I'd like to study the relationship between a circle and the hexagon surrounding it (the circle is inscribed in the hexagon). The diagram below shows three such hexagons.  The circles have radius r, and the distance from the centre of the circle (O) to the corner of the hexagon is h (the hypotenuse of the triangle formed by the radius and the tangent).  The angle between the radius and the hypotenuse is 30 degrees.

Given that r = 1 unit, what's the length h?

Trigonometry tells us that cos 30 = r/h therefore h = r / cos 30 = r / 0.866
h = 1.154 r

This means that the additional distance ( h - r ) = 0.154 r or 15% of the radius of the larger circle.  This additional distance is the maximum radius of a circle that would fit in the space between the three circles shown in the diagram.  For example, if the larger circles have a radius of 1 metre, the radius of the smaller circles would be 0.15m or 15 cm.  As I've calculated previously, there isn't much space between hexagonally-packed circles.


Atoms in metallic solids are typically hexagonally packed, and alloys are formed when other atoms are mixed with a pure metal.  It occurred to me that it may be possible for these atoms to be small enough to fit into the gaps between the atoms - using the mechanism I've indicated above.


Here are the radii of some metal atoms, for reference, in picometres.  1 picometre, = 1 x 10 -12 metres

Fe (iron)  156
Sn (tin)  145
Al (aluminium) 118
Co (cobalt) 152

15% of these values gives us 24, 21, 17, and 22.    The atomic radius of carbon (which is alloyed with iron to make steel) is 67 pm.  So, there is no way that carbon will fit neatly into the gaps in the matrix of hexagonally-packed iron atoms.   [Indeed, helium has the lowest atomic radius, and that's 31 pm, still too large].

Instead, in interstitial alloys, the smaller atoms in an alloy distort the close-packed arrangement of the metal and that's what affects its physical properties.

Further reading

Previous post on the radius of a circle in the corner of a circle
Space filling calculation 3D - sphere in cube, sphere in hexagonal prism
Hexagonal close packing - 2D-filling calculation

 

Geometry: A Circle in the corner of a circle

This article is specifically written to answer the geometry question:  "What is the radius of a circle drawn in the space between a circle of radius 1 unit, and the corner of the enclosing square?" To better explain the question, and then answer it, I have drawn the diagram shown below.  The question states that the radius of the larger circle is 1 unit of length, and that it is enclosed in a square. What is the radius of the smaller circle drawn in the space in the corner region?  (The question was asked in a GCSE workbook, aimed for children aged 14-16, although the geometry and algebra became more complicated than I had expected).

The diagram isn't perfect, but I'm better at using a pencil and compasses than I am with drawing geometric shapes in Photoshop.  The larger circle has centre E, and DE = EF = 1 unit.  What is the radius of the smaller circle, with centre C (BC = CD = smaller radius).

By symmetry, the angle at A = 45 degrees, so triangles ACG and AEH are right-angled isosceles triangles.
AH and EH are equal to the radius of the larger circle = 1 unit
By Pythagoras' Theorem, length AE = √2

Length AD = AE - DE = √2 -1
However, length AD is not the diameter of the smaller circle.  The diameter of the smaller circle is BD, not AD.  We are still making progress, nonetheless.

Next, consider the ratio of the lengths AD:AF.
AD = √2 -1 as we showed earlier
AF = AD + DF = (√2 -1) + 2 (the diameter of the circle) = √2 + 1

So the ratio AD:AF =  √2 -1 :  √2 + 1

And the fraction AD/AF = √2 -1 /  √2 + 1

What is the remaining distance between the circle and the origin?

Look again at the larger circle, and the ratio of the diameter to the distance from the corner to the furthest point on the circle?

The fraction AB/AD is equal to the fraction AD/AF.  This fraction describes the relationship between the diameter of a circle and the additional distance to the corner of the enclosing square.  The diameter of the circle is not important, the ratio is fixed.

So we can divide the shorter length AD in the ratio AD:AF, and this will give us the length AB and (as we already know AD) the diameter of the smaller circle, BD.

To express it more simply and mathematically:  AD/AF = AB/AD
Substituting known values for AD and AF, this gives:

AD/AF = AB/AD

(AD^2) / AF = AB

(√2 -1)^2 /  (√2 + 1) = AB


 Evaluating:
(√2 -1)^2 = 3 - 2 √2 = 0.17157...

and:
(√2 + 1) =  2.4142...

Now:  BD (diameter or circle) = AD ('corner' of larger circle) - AB ('corner' of small circle)
Substituting values for AD and AB, and then combining terms over the same denominator, we get:

 Having combined all terms over the same denominator, we can now simplify (√2 -1)(√2 +1), since (a+b)(a-b) = a^2 - b^2

BD is the diameter of the smaller circle, BD = 0.216...  Comparing this with the diameter of the larger circle, which is 2.00, we can see that the smaller circle is around 10% of the diameter of the larger one.  This surprised me - I thought it was larger.

In future posts, I'll look at other arrangements of circles in corners - in particular the quarter-circle in the corner (which, as a repeating pattern, would lead to a smaller circle touching four larger circles in a square-like arrangement), and a third-of-a-circle in the corner of a hexagon.  I'll then compare the two arrangements (in terms of space filled) and also check against any known alloys, looking at the ratios of diameters to see if I can find a real-life application.

Other 'circle' posts:

A Circle in the Corner of a Circle
A Circle in the Corner of a Hexagon
Close-packed Circles - calculating the space occupied

Close-packed Spheres - calculating the volume occupied

Pitfalls of Online Optimisation and Testing 3: Discontinuous Testing

Some forms of online testing are easy to set up, easy to measure and easy to interpret.  The results from one test point clearly to the next iteration, and you know exactly what's next.  For example, if you're testing the height of a banner on a page, or the size of the text that you use on your page headlines, there's a clear continuous scale from 'small' to 'medium' to 'large' to 'very large'.  You can even quantify it, in terms of pixel dimensions.  With careful testing, you can identify the optimum size for a banner, or text, or whatever it may be.  I would describe this as continuous testing, and it lends itself perfectly to iterative testing.

Some testing - in fact, most testing - is not continuous.  You could call it discrete testing, or digital testing, but I think I would call it discontinuous testing.

For example:
colours (red vs green vs black vs blue vs orange...)
title wording ("Product information" vs "Product details" vs "Details" vs "Product specification")
imagery (man vs woman vs family vs product vs product-with-family vs product alone)

Both forms of testing are, of course, perfectly valid.  The pitfall comes when trying to iterate on discontinuous tests, or trying to present results, analysis and recommendations to management.  The two forms can become confused, and unless you have a strong clear understanding of what you were testing in the first place - and WHY you tested it - you can get sidetracked into a testing dead-end. 

For example; let's say that you're testing how to show product images on your site.  There are countless ways of doing this, but let's take televisions as an example.  On top right is an image borrowed from the Argos website; below right is one from Currys/PC World. The televisions are different, but that's not relevant here; I'm just borrowing the screenfills and highlighting them as the main variable.  In 'real life' we'd test the screenfills on the same product.

Here's the basis of a straightforward A/B test - on the left, "City at Night" and on the right, "Winter Scene".  Which wins? Let's suppose for the sake of argument that the success metrics is click-through rate, and "City at Night" wins.  How would you iterate on that result, and go for an even better winner?  It's not obvious, is it?  There are too many jumps between the two recipes - it's discontinuous, with no gradual change from city to forest.

The important thing here (I would suggest) is to think beforehand about why one image is likely to do better than the other, so that when you come to analyse the results, you can go back to your original ideas and determine why one image won and the other lost.  In plain English:  if you're testing "City at Night" vs "Winter Scene", then you may propose that "Winter Scene" will win because it's a natural landscape vs an urban one.  Or perhaps "City at Night" is going to win because it showcases a wider range of colours.  Setting out an idea beforehand will at least give you some guidance on how to continue.

However, this kind of testing is inherently complex - there are a number of reasons why "City at Night" might win:
- more colours shown on screen
- showing a city line is more high-tech than a nature scene
- stronger feeling of warmth compared to the frozen (or should that be Frozen) scene

In fact, it's starting to feel like a two-recipe multi-variate test; our training in scientific testing says, "Change one thing at a time!" and yet in two images we're changing a large number of variables.  How can we unpick this mess?

I would recommend testing at least two or three test recipes against control, to help you triangulate and narrow down the possible reasons why one recipe wins and another loses. 

Shown on the right are two possible examples for a third and fourth recipe which might start to narrow down the reasons, and increase the strength of your hypothesis.  
 
 If the hypothesis is that "City at Night" did better because it was an urban scene instead of a natural scene, then "City in Daylight" (top right) may do even better.  This has to be discontinuous testing - it's not possible to test the various levels of urbanisation; we have to test various steps along the way in isolation.

Alternatively, if "City at Night" did better because it showcased more colours, then perhaps "Mountain View" would do better - and if "Mountain View" outperforms "Winter Scene", where the main difference is the apparent temperature of the scene (warm vs cold), then warmer scenes do better, and a follow-up would be a view of a Caribbean holiday resort. And there you have it - perhaps without immediately realising, the test results are now pointing towards an iteration with further potential winners. 

By selecting the test recipes carefully and thoughtfully and deliberately aiming for specific changes between them, it's possible to start to quantify areas which were previously qualitative.  Here, for example, we've decided to focus (or at least try to focus) on the type of scene (natural vs urban) and on the 'warmth' of the picture, and set out a scale from frozen to warm, and from very natural to very urban.  Here's how a sketch diagram might look:

Selecting the images and plotting them in this way gives us a sense of direction for future testing.  If the city scenes both outperform the natural views, then try another urban scene which - for example - has people walking on a busy city street.  Try another recipe set in a park area - medium population density - just to check the original theory.  Alternatively, if the city scenes both perform similarly, but the mountain view is better than the winter scene (as I mentioned earlier), then try an even warmer scene - palm trees and a tropical view.

If they all perform exactly similarly, then it's time to try a different set of axes (temperature and population density don't seem to be important here, so it's time to start brainstorming... perhaps pictures of people and racing cars are worth testing?).

Let's take another example:  on-page text.  How much text is too much text, and what should you say? How should you greet users, what headlines should you use?  Should you have lengthy paragraphs discussing your product's features, or should you keep it short and concise - bullet points with the product's main specifications?

Which is better, A or B?  And (most importantly) - why?  (Blurb borrowed and adapted from Jewson Tools)

A: 
Cordless drills give you complete flexibility without compromising on power or performance.  We have a fantastic range, from leading brands such as AEG, DeWalt, Bosch, Milwaukee and Makita.  This extensive selection includes tools with various features including adjustable torque, variable speeds and impact and hammer settings. We also sell high quality cordless sets that include a variety of tools such as drills, circular saws, jigsaws and impact drivers. Our trained staff in our branches nationwide can offer expert technical advice on choosing the right cordless drill or cordless set for you.

B:
* Cordless drills give you complete flexibility without compromising on power or performance.
* We stock AEG, DeWalt, Bosch, Milwaukee and Makita
* Selection includes drills with adjustable torque, variable speeds, impact and hammer settings
* We also stock drills, circular saws, jigsaws and impact drivers
* Trained staff in all our stores, nationwide


If A was to win, would it because of its readability?  Is B too short and abrupt?  Let's add a recipe C and triangulate again:

C:
* Cordless drills - complete flexibility
* Uncompromised performance with power
* We stock AEG, DeWalt, Bosch, Milwaukee and Makita
* Features include adjustable torque, variable speed, impact and hammer settings
* We stock a full range of power tools
* Nationwide branches with trained staff

 C is now extremely short - reduced to sub-sentence bullet points.  By isolating one variable (the length of the text) we can hope to identify which is best - and why.  If C wins, then it's time to start reducing the length of your copy.  Alternatively, if A, B and C perform equally well, then it's time to take a different direction.  Each recipe here has the same content and the same tone-of-voice (it just says less in B and C); so perhaps it's time to add content and start to test less versus more.


D:
* Cordless drills - complete flexibility with great value
* Uncompromised performance with power
* We stock AEG, DeWalt, Bosch, Milwaukee and Makita
* Features include adjustable torque, variable speed, impact and hammer settings
* We stock a full range of power tools to suit every budget
* Nationwide branches with trained and qualified staff to help you choose the best product
* Full 30-day warranty
* Free in-store training workshop  

E: 
* Cordless drills provide complete flexibility
* Uncompromised performance
* We stock all makes
* Wide range of features

* Nationwide branches with trained staff

In recipe D, the copy has been extended to include 'great value'; 'suit every budget', training and warranty information - the hypothesis would be that more is more, and that customers want this kind of after-sales support.  Maybe they aren't - maybe your customers are complete experts in power tools, in which case you'll see flat or negative performance.  In Recipe E, the copy has been cut to the minimum - are readers engaging with your text, or is it just there to provide context to the product imagery?  Do they already know what cordless drills are, what they do, and are they just buying another one for their team?

So, to sum up:  it's possible to apply scientific and logical thinking to discontinuous testing - the grey areas of optimisation.  I'll go for a Recipe C/E approach to my suggestions:

*  Plan ahead - identify variables (list them all)
*  Isolate variables as much as possible and test one or two
*  Identify the differences between recipes 
*  Draw up a continuum on one or two axes, and plot your recipes on it
*  Think about why a recipe might win, and add another recipe to test this theory (look at the continuum)

The  articles in the Pitfalls of Online Optimisation and Testing series

Article 1:  Are your results really flat?
Article 2: So your results really are flat - why?  
Article 3: Discontinuous Testing