A man had a room with only one window in it.  It was a square window, 1 meter high and 1 meter wide (see diagram).  Unfortunately, the man had sensitive eyes and the window let in too much light. The man sent for a builder, and told him to alter the window so that it let in half the amount of light.  To make the matter especially tricky, the man insisted that the resulting window also had to be square, and 1 meter high and 1 meter wide.  The builder isn’t allowed to use curtains, or shutters, or coloured glass. How did the builder solve the problem?

If you have not tried to solve it, have a go now.  For everyone else, the answer is after the break.

The builder did this…..

Did you solve it?  Any other answers?

I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for theKindle (UK here and USA here) and on the iBookstore (UK here in the USA here). You can try 101 of the puzzles for free here.

1. Gib says:

Tricky…. I thought of angling the window by 60 degrees, so it’s half inside the house, half outside like some abomination….. But even then it depends on where the light comes from as to how much actually gets in.

I didn’t read the problem as analy as I should have…

1. Stephen says:

I had this answer as well, but imagined that the whole house was rotated with it. That does require that the window is perpendicular to the line of sight with the Sun initially though (though other rotations would work if this is not the case)

2. Anon says:

The builder put four triangular pieces of blackout tape on the window?

4. Dave2 says:

I have an easier solution. Knock the house down and rebuild it on a space station twice the distance from the sun.

1. Edd says:

That’d give you a quarter of the light – it follows an inverse square law.

2. Edd says:

That’d give you a quarter of the amount of light – it follows an inverse square law.

3. Eds don’t be daft.

4. Matthew, why is Edd being daft?

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5. Didn’t realise Edd’s comment was referring to Dave2’s. Thought it was referring to the given solution..

Mea Culpa, sorry. Looks like I’m the daft one.

6. Dave2 says:

You are indeed correct. I’m the Chief Mathematician at the CERN Large Hadron Collider. I better go back and check my workings out.

7. dharmaruci says:

quarter of the light they say. but if spinning/rotating to always face the Sun, then quarter of light for 100% of time (day and night) is half the original light on averrage – so answer works,

8. Rich says:

And that would be easier would it? haha.

5. I would love to meet the “builder” who could work that one out.

1. Anonymous says:

That’s a bit anti-working class, Russ.

2. Ed Milliband says:

Where does Russ mention class and what’s wrong with being anti-working class anyway?

6. Kristian says:

Clever. I couldn’t see how reducing the window dimensions to 0.707m (square root of 0.5) would be a wrong answer, so I just went with that. But I like this one more: No measuring is needed, just the half points of the sides.

1. That in fact is what’s done here – calculate the lengths of the sides and you’ll see that they’re all 0.707m long. The trick is that the window has to be 1m high and 1m wide; if you only reduce the dimensions, you can’t accomplish this. However, if you rotate the window as well as reducing the dimensions, as the solution shows, you can accomplish this.

7. solver says:

This won’t assure you to have half of the light – just because a strength of light can be different in different places of window. Just imagine a bird, that would fly in front of the window. It could be whole in unshaded area resulting in less than 1/2 of light or in shaded area resulting in more than one half of light.

Only my solution was polarisation filter. It guarantees exactly 1/2 of unpolarised light, and it is not a curtain, nor shutter nor colored glass.

1. Yep I had a number of solutions including Richards and a polarisation filter. Bonus point for ne, I also predicted which answer Richard would provide.Question though, which way did you orient your filter. Hint “it doesn’t matter” is not a correct answer. I went for 45 degrees, meaning that it would filter out half the unpolarised, but also half(ish) the polarised light reflected off any horizontal or vertical surface.

8. Gib says:

Remember though that this result lets in half the power of light (or number of photons). Our eyes/brain wouldn’t interpret this as half the light though, because they’re not linear, they’re more logarithmic.

9. Christian says:

How about just adding depth to the window – it will have the exact same dimensions as before, but only allow half the amount of light to enter. It might look strange from the outside, but the question didn’t say anything about aesthetics. Something like http://imgur.com/yaQRF

1. Engywuck says:

This was indeed the first solution I came up with too. Find my second/Richard’s solution more elegant.

10. I thought I had an answer which was “obvious” but I didn’t think of this one!

11. Lazy T says:

That’ how I’d get 50% less but up here in the North I’d like to get 50% more

1. mittfh says:

If only… 🙂

Now I know that was said at least partly in jest, but if you decided to set it as a problem while following the same constraints as the original problem, it would be quite tricky to achieve (I assume getting the builder to uproot your house and garden and move them a few hundred miles south, or even uprooting it and rotating it so said window faces south is out of the question!).

Erm, add a lens to the exterior? Add an array of mirrors in the garden?

2. Lazy T says:

’twas purely in jest, but now I’ve just wasted 20min only to conclude that I can only get 50% more, whilst leaving the window 1×1, by putting in another window

12. Gareth says:

While I did get the correct answer, I also considered the idea that the builder, annoyed at being given such a challenge from an awkward customer, left the window as it was and poked one of the customer’s eyes out. Thus the window retains its 1m x 1m dimensions, and the customer’s blindness in one eye means only half the amount of light gets in.

1. Classy says:

Your answer is by far superior to the rest!! Simple, yet effective!

13. How about sticking a square of blackout material size 707cm on a side onto the middle of the window? Half a square metre of the window would be covered.

14. The laziest builder would wait for a dull rainy day and then turn up for his money, or offer the man sunglasses, or move the house to another planet, or dim the sun, or paint the floor and inside of the house with soot to minimise back-reflection, or give the man eye surgery so that he could cope with a normal window, or change architectural fashion so that rectangles were the new trend, or…

15. As I said on Friday, this is interesting as the right answer is not right — that is, the orientation of the new window is not perpendicular!

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1. Don’t see anything in the puzzle specifying that the new window has to be ‘perpendicular’

2. The right answer is right (correct) it’s just not right (perpendicular).

And that’s just not normal. 😉

3. Neil says:

He’s being clever (‘right’ equalling ‘perpendicular’), letting you know that he mentioned the solution on Friday… someone else who can’t bear not to hint at the answer even when asked not to.

4. @ Jass : Quite.

@ Matthew : Now you’re plumbing the depths… 😀

@ Neil : Please show me where Richard asked people not to hint at the right answer… In any case, my memory is imperfect: I had thought to say, “the right answer is not right”, but resisted the temptation as it might have been too much of a clue. I settled for an obscure reference to a “Schrödinger’s answer”.

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16. This is the answer I came up with (and even tried it in photoshop). But…doesn’t this shape get called a diamond now? I know it still is a square but the angle you view it at now makes it a diamond for descriptive purposes.

17. Virgil says:

Do nothing to the window, but tilt the wall at an angle of 30 degrees to horizontal. Assuming the light source is not on the ground, shouldn’t this reduce incoming light by about one half?

18. Isn’t this new window √2/2-meter-high?

1. Niva says:

If “high” means the vertical measurement, then it’s 1 meter. √2/2 is the length of each side of the inner square, so its area is 1/2 as required. I got the same answer as Richard.

2. david says:

The height of a square is the length of one of its sides – it’s an abstract shape and that’s how we measure them. Richard’s answer is strictly incorrect.

19. Surely the resulting window is no longer a 1 meter high and 1 meter wide square, but either a 1×1 rhombus or a 0.707×0.707 square.

20. M says:

I thought: thicken the wall. It leaves a lot less light through. How much the wall needs to be thicker, I did not calculate…

I did not get the real answer.

1. spiderabc1 says:

haha

2. spiderabc1 says:

That wasn’t for you getting it so called wrong. Just a funny.

21. silenttex says:

Surely the simplest solution is to build a wall directly outside the window, to cover half the window? Window dimensions are unchanged, half the light is blocked.

22. dpettus says:

Plant a tree.

23. Jerry says:

Very clever.

24. AL says:

The Comments here read like a Mad Hatters tea party. What is not “perpendicular” about the new window? Why is it not still 1 meter high and 1 meter wide? It is both.

Wake up the dormouse.

1. Anon says:

In answer to your question about height and width – these terms are usally applied to an object independent of its orientation. RW has used these terms to apply to a square in two different ways dependent on its positioning relative to the (notional only)perendicular.
I trust that this helps your understanding.

As is often the case

25. To be honest I was more inclined to the smacking him in the eye solution. What sort of idiot puts those sorts of constraints on a builder?

26. The other Matt says:

Dont got it….
But its not correct to change the “rules” about measuring !

27. Neil says:

I had the window built into a corner so half was on one wall, half on the other. Didn’t really take the time to work out if that would actually cause less light to come in.

28. david says:

I have to say I don’t understand this answer. The window has black bits around it and is no longer the same dimensions.

Oh, I see what you’ve done. It’s verbal sophistry. Confusing height above sea level with height of a geometric shape. I say this is a poor answer. My answer is to locate the window lower in the wall using maths with angles and the distance to the sun.

1. Anon says:

Sophistry it is.
Spot on.

29. spiderabc1 says:

I had my main answer (an hour glass type effect; 1m across the top, in at the middle and out again, 1 m high.) The second a diamond, like shown. Then I saw an entry on here about a square shape. I was considering having the middle blanked out over the sides because of natural sunlight brightness mid-day. This one made sense to me.

30. One Eyed Jack says:

I’m surprised by how many people didn’t see this answer coming. I’m not knocking anyone’s cognitive ability. I’m just surprised because it’s an old puzzle that pops up frequently.

31. One Eyed Jack says:

One another note (in the spirit of unusual solutions), did anyone mention silvering the glass? Perhaps this falls under the “colouring” restriction. 🙂

1. Anon says:

Since this puzzle hinges on the precise meaning of words (or meanings of words) my reading of the question is that this is a permissable answer – in that silvering is not specifically excluded.
It was one of my solutions

32. Rudi says:

My solution was similar but instead I thought of a blackened border.

I also thought he could have replaced the old window with a pane of polarised glass!

33. Anonymous says:

“Anon Says:
April 30, 2012 at 7:22 pm
In answer to your question about height and width – these terms are usally applied to an object independent of its orientation. RW has used these terms to apply to a square in two different ways dependent on its positioning relative to the (notional only)perendicular.
I trust that this helps your understanding.

As is often the case”

What? This is a Mad Hatter party! The darn diamond is 1 meter high and 1 meter wide, top to bottom, and left to right. What tells you otherwise? RW neatly solved his conundrum by turning the window into a diamond shape which is half the area of the square. The height and width are 1 m now regardless “of the object’s orientation”. Its area is half the square (according to RW, I assume the geometry is correct, which it seems to be just using one’s eyes – the black triangles take away half the area). The perpendicular is not notional. It is a fixed element of the puzzle, as a wall is naturally perpendicular to the ground.

These comments get odder and odder. One begins to see how Lewis Carroll dreamed up the idea of passing through the looking glass.

1. david says:

Richard’s answer breaks the implicit rules every bit as much as silvered glass would. A diamond is only a square in abstract, but in abstract its height is the length of one of its sides. The new window is 1m high only when it isn’t square. Imagine a 1m square in three dimensions, not on the wall but flat on the carpet. Now slide it into the wall it becomes a slit. Can the builder call that a square too? “But you aren’t imagining extra dimensions!” He could protest, while Alice giggled.

2. Anon says:

Anonymous

The perpendicular is notional because at no point in the puzzle does RW tell us which direction is perpendicular. We infer where the perpendicular is from our normal interpretation of drawings such as the one RW made.

RW draws a square with the sides parallel to our computer screens. We assume, by convention, that the line of the square at the bottom of the screen is the lower one and is parallel to the ground. We also assume that the two sides adjacent to this one are upright and perpendicular the the ground. These are only assumptions based on our conventions however. To make the puzzle “work” we have to adhere to this convention.

Paradoxically to make it “work” we also have to disregard the convention that the height of square is the term used to denote the length of its side.

I trust that this has helped your understanding of this puzzle, but if you seek further clarification please do not hesitate to ask.

BTW Everyone at this table got RW’s given answer in under a minute

34. Chip says:

I appreciate why the ‘right’ answer is what it is. But a nice awning could also be correct, stylish, and easier to install.

35. Dormouse says:

More than one Mad Hatter at the Tea party! Lewis Carroll must be rolling in his grave – out of frustration that he can’t use this stuff in a new book.

Mad Hatter David writes: “Richard’s answer breaks the implicit rules every bit as much as silvered glass would. A diamond is only a square in abstract, but in abstract its height is the length of one of its sides.”

What goobledegook of genius is this? The height of a diamond or any other shaped window is the distance from its bottommost edge to its topmost edge, when it is viewed by an eye along a horizontal plane parallel to the ground.

The window could be squiggly in shape and still its height would be bottommost part of its bottom edge to the topmost part of its top edge. But maybe the sun doesn’t rise in the East.

“The new window is 1m high only when it isn’t square. Imagine a 1m square in three dimensions, not on the wall but flat on the carpet. Now slide it into the wall it becomes a slit. Can the builder call that a square too? “But you aren’t imagining extra dimensions!” He could protest, while Alice giggled.”

Phone home Lewis Carroll! May one ask, did you get out of a bed this morning? Would you call that bed a King size, a Queen size or a Slit? Did it stop being 8ftx 11ft because it was horizontal? Is the bed a pull down bed which was originally 11ft tall when it was a hideaway in the wall but turned into a Slit when it was horizontal, a Slit only 1ft high? If so how did you get into it? Did you have to slide yourself in? Did your feet fit?

Mad Hatter Anon Says: “The perpendicular is notional because at no point in the puzzle does RW tell us which direction is perpendicular. We infer where the perpendicular is from our normal interpretation of drawings such as the one RW made.”

We infer the window is vertical because no one mentioned an attic. Or any other room where a window would not be simply vertical, as 99% of windows are.

“RW draws a square with the sides parallel to our computer screens. We assume, by convention, that the line of the square at the bottom of the screen is the lower one and is parallel to the ground. We also assume that the two sides adjacent to this one are upright and perpendicular the the ground. These are only assumptions based on our conventions however. To make the puzzle “work” we have to adhere to this convention.”

We make the puzzle work by assuming that it is a window in a wall of a room, as RW stated, with all the normal attributes of a window. It has nothing to do with a computer screen. He could have posed the same question verbally. We assume a wall is vertical and perpendicular to the ground because unless we are told otherwise we accept the norm as the meaning intended.

“Paradoxically to make it “work” we also have to disregard the convention that the height of (a) square is the term used to denote the length of its side.”

The convention is that the height of a square or any other shape including a statue of David by Michelangelo is the distance of its topmost edge from its bottommost edge relative to the center of the Earth. There is no convention that the height of a square in the length of its side. It depends on whether the square is placed firmly on the ground with one side going along the ground, or if it is tilted.

The whole point of RW’s puzzle is that most people find it hard to think of tilting the square, partly because then the diagonal becomes the height and it is not actually the same one meter. It is longer. Similarly, the width of the diamond window is now also the diagonal, which is also longer.

But according to the puzzle the height and width still have to be one meter. So reducing the height and the width back to one meter each is what actually makes the square smaller. So it is less than the size of window it was. It is a diamond whose height is one meter and whose breadth is one meter. That makes it more than half of what it was, however. It is somewhere in between.

Pythagoras came up with the formula a_2 + b_2= c_2 for the sides of a right angled triangle, if my high school maths memory is correct. So the original square window with horizontal and vertical edges is one original side squared, a_2, ie 1 meter square, which is 1 square meter.

What is the area of the diamond then? Its bisecting vertical diagonal is one meter, therefore its sides are smaller, a and b are now somewhat smaller. But you can see from the geometry how much the area is reduced – by one half.

So the answer you Mad Hatters came up with in less than one minute, the answer provided by RW, is immediately available if you tilt the window. Congratulations for thinking of such a novelty so fast!

But even if you did come up with it in a minute, coming up with it in a minute would prove nothing either way. It is just as easy to come up with a wrong answer in a minute. Actually, probably much easier.

So what is really in order is to congratulate you on your powers of wacky thinking.

But as the above comments prove, you can carry wackiness too far. You should bathe in your glory without revealing just how wacky your thinking was to reach the right answer – or at least, conceal any further extrapolations of it into the realm of the superwacky, as above.

One reason is because it completely stymies the thinking of the Dormouse. I cant solve the puzzle of how long the sides of the diamond are! 1= a_2 + b_2 but a=b so 1= 2(a_2) … so a= Square root of 1/2?

Surely not?!

Back to tea.

1. david says:

“It is a diamond whose height is one meter and whose breadth is one meter”

Yes. Or a square whose height and breadth are sqrt 1/2. Therefore if the height of the window is 1m it is a diamond, not a square.

2. david says:

To clarify: squares no more exist in the physical world than perfect circles do. They’re platonic. We measure them by the length of their side. Windows exist in the physical world and their height is measured in relation to the ground (or gravity – not sure which would feel more wrong in a slanty building). Hence I call shenanigans in conflating the two. It’s a trick, a piece of misdirection. Which is totally legitimate from a magician, but is a solution of a different nature to, for example, the one for the famous “draw 4 lines to connect 9 dots in a square” puzzle. Worth recognising this difference.

3. Alice says:

Dear Dormouse/Anonymouse

I thought it sensible to consult Mr Carroll, who thought up this puzzle in the first place. He reported as follows:-

1. Mr. Mad Dave defines the height of a diamond in relation to its sides. Mr Dormouse defines height in relation to the “ground.” On balance, I am afraid, he preferred Mr. Mad Dave’s definition, because there is not always a ground around to define things by, but there are always sides to a diamond. Presumably, he said, that a diamond in outer space has no height on this basis.

2. He did, respectfully, ask me to point out to you that at no point in the puzzle does Mr Wiseman say where the “ground” is, so presumably you could not solve his puzzle for this reason.

3. Finally he did say that if he had known all the fuss it would cause he might not have devised the puzzle in the first place. “Get a life” was the actual phrase he used.

I remain, sir, your obedient servant

Alice

36. ivan says:

I don’t think the customer will consider that this meets the spirit of his intentions. But then the spirit of his intentions may not have been satisfiable.

37. Dormouse says:

Um.. not sure where the conflation and misdirection comes in. The fact that a square or a diamond shape exists in our imaginations as a “Platonic ideal” doesn’t mean that objects in the real world cannot fit those abstract templates, does it? A window can be a true square in real life, and windows typically are vertical in relation to the center of gravity of the Earth, so that we can all lie in our beds without slipping off, no? Not sure where the magical conflation comes in.

Your ref to 4 lines connecting 9 dots in a square reminds me that RW posed the question how does one line connect the same 9 dots, isn’t that right? Maybe I misremember. I saw that somewhere. Forgot to check the answer when it baffled me.
But that must have been conflation and misdirection, surely!

1. david says:

A platonic square is measured by the length of its side. A platonic diamond is a square and I don’t know what’s formally correct but it would be reasonable if you wanted to say its height was the square’s diagonal. This window is a square and its height is 1m, but not at the same time. So, conflation.

2. Dormouse says:

ALice says : “he preferred Mr. Mad Dave’s definition, because there is not always a ground around to define things by, but there are always sides to a diamond. Presumably, he said, that a diamond in outer space has no height on this basis.”

I believe you may not have been speaking to the same Lewis Carroll that I called. He told me that as long as anything was measured along a line extruded from the center of gravity of the Earth that would be its “height”. Certainly it would be if it was in space. If it was on Mars, same thing, just have to be measured on a line starting from the center of gravity of Mars. The measurement would be the same. It would be the same wherever you started the line, as a matter of fact. But the object would have to be oriented in a specific way to get its height for that orientation. In the case of the window, of course it is oriented by the wall, which unless it is the side of a roof attic, would be vertical to the center of gravity of the Earth.

None of this needs any reference to the ground.

Alice also says : “Finally he did say that if he had known all the fuss it would cause he might not have devised the puzzle in the first place. “Get a life” was the actual phrase he used.”

Yes, you must have been speaking to the wrong Lewis Carroll. Only Cambridge dons would be so democratically impertinent to a stranger. Oxford dons are always kind from on high and would never hurt the feelings of a shut in who has nothing better to do than comment on RW’s blog puzzles. Lewis Caroll himself the real Carroll told me that in fact he could think of no better role in life than to comment on puzzles. In fact his two books were precisely that. He looks forward to similar books from the Mad Hatters here.