# Answer to the Friday Puzzle!

121

On Friday I set this puzzle…

Imagine the Earth is perfectly circular and has a circumference of 24,000 miles.  If you had a piece of rope stretched around it, touching the floor, that rope would also be 24,000 miles long. If you were to add 3 feet to that rope and then make it hover above the ground until it’s a perfect circle again, approximately how far off the ground would it get?

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

The rope would raise 6 inches off the ground if you added 3 feet to it! Can anyone explain why and what is surprising about this answer?
I have produced an ebook containing 101 of the previous Friday Puzzles! It is called PUZZLED and is available for the Kindle (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.

Advertisements

## 121 comments on “Answer to the Friday Puzzle!”

1. Anne Elk says:

I didn’t and don’t have a clue about how to do this one.

• Barry Goddard says:

I have not now posted for over two weeks. Not since the name calling and other unpleasantness reached for me an unacceptable level.

Yet I see in the past two weeks more and more posts being unpleasant and rude towards me. It is now clear where the problem lies. It lies with the people who are using this blog to vent their splene rather than make civilised discussion.

As to the puzzle at hand. There is an easy visualisation. Imagine the new three foot length rolled into its own circle. It would have a radius of about six inches.

Now imagine the big circle being slightly elastic and streched so the new little circle can slip between it and the Earth. The little circle would be holding the big circle 6″ off the ground. And that is the answer needed.

Puzzles like this do not need complicated maths or similteneous equations. Simple visualisation and intuition will result in the right answer at an instinctive level.

Because even before we had language and maths and the need to write everything down we could dodge thrown rocks and hide from tigers and make fire. We have these innate abilities. Do not let others undermine you because they have “clever” learning methods.

I have yet to get one of these puzzles wrong yet all I do is quietly meditate for a few moments until my mind is attuned with the solution.

• Eddie says:

In para 4 you say ‘that is the answer needed’. But why is that the answer?

• Slange Navarr says:

I missed you, Barry. Thought you wouldn’t be coming today. Good to see you. You always make me smile

• Ken Haley says:

Barry, I’m surprised you didn’t point out the variability of the possible answers to this puzzle given the obvious elasticity of any rope, and the fact that we weren’t told how tightly the rope was stretched, nor the effects of the various temperatures and pressures that the rope would endure as it made its way through the varying weather conditions. I think your analysis is slipping. 😉

• DAve (not 22/2) says:

Surely the thickness of the rope needs to be taken into account too? Would 1 cm thick rope be higher off the ground than say a 4 cm thick rope?

• Simon says:

You must experience time differently from everyone else, Barry. Your previous comment was last Wednesday, 16 July at 5:01 PM which, by my calculations was less than 5 days before you made this one, not “over two weeks”.

You’re confusing radius and diameter again. If you let a circle with a 6 inch radius slip between the big circle and the Earth, it will lift the big circle 12 inches off the ground.

Your last sentence has already been shown by many people to be false.

• John D. says:

@Barry
I don’t know how you can easily visualize that a 3 foot rope rolled into a ball would have a radius of 6 inches. I have to do some math to determine that radius. When I do use math I find the actual answer to be 5.7 inches. Maybe you can simply visualize that but not everyone can. Don’t disparage people because they can’t visualize what you can.

Then of course if you did slip this circle of 6 inch radius between the earth and the rope, it would rise by twice the radius. So it appears you are wrong and your meditation and visualization didn’t quite help you out here.

I’ll have to resort to my math, which wasn’t all that complex and I didn’t need simultaneous equations either.

• Anonymous says:

I think you’ll find it’s *similtaneous* equations (according to Bazzer anyway).

• Sydious says:

I see what Barry means : if you consider the extended rope forming a circle, but touching the Earth in one point, then at the other end of the diametre, the rope is above the Earth, with a distance equal to the difference of the two diametres (D=d(rope)-d(Earth)).
This difference of diametres is twice the distance that the rope would hover above Earth if it hovered uniformly, and, since the diametre of a circle is proportional to its circumference, it is also the diametre of the circle formed by the additional length of rope. So if the additional circle has a 12″ diametre, then the rope would hover 6″ above the Earth.
(Sorry if my explanation is a little confuse, I’m not a native english speaker. And I don’t even know if I could explain this clearly in my own language ^^)

• Self Appointed Guardian says:

That’s “spleen”, Barry.

You may be an instinctive problem solving genius, but your spelling is terrible.

• Phil McCavity says:

Its “you’re” not “your”

• Ben Down says:

Phil, if *you’re* referring to the comment immediately above *yours* then I’m afraid *you’re* mistaken.

• ChrisR says:

and Phil – your “its” should be “it’s”. (“its” is possessive, “it’s” is short for it is)
But perhaps you meant that and realised only after clicking ‘Post Comment’.
Lack of an edit option (unless you’re Richard and change the question 😉 ) can lead to unwanted embarrassment but does ensure all our ramblings remain unchanged for posterity – which perhaps has some benefits

• Barry Goddard says:

@Sydious

Of the many who come to sneer one understands. If only the “mathematically literate” who are stuck in their ways could be as open to intuitive insight.

Even those who point out they can solve this puzzle by similteneous equations (one for each rope) have much to learn from a more holistic approach. I am happy to teach those who wish to learn.

• Starman says:

The one who sneers yet again fails to understand. If only this mathematically illiterate person who is stuck in his ways could be open to introspection.

No-one has claimed to solve this puzzle by similteneous (or simultaneous) equations, that is another fabrication on your part. I am happy to teach you but you have made it clear that you have no wish to learn.

• Phil McCavity says:

… and the bait is taken.

• Eddie says:

So Barry’s just taking the piss is he? Thought so.

• Anonymous says:

Troll / not troll, he’s still being a dick. There’s no justification for him to behave like that. Who does he think he is?

• Barry Goddard says:

@Self Appointed Guardian

Thank you.

It is humbling to be called an instinctive problem solving genius.

I believe my talents are humble and available to all.

Yet one way to measure a human mind is to measure the contributions it makes to society. If my humble contributions be recognised as genius that is all I need to feel welcomed and humbled.

Some weeks ago @Steve called me a godlike being. That too was high praise. Perhaps too high. Yet it was his authentic response and I welcome that.

Yet any one of us can learn my skills. Just follow what I do and do the same. It is my pleasure to help you all.

• Andy Kaufman says:

You’re brilliant, Barry, a genuinely talented performance artist.

You’re better than I ever was.

• DAve says:

All hail to the demi-god that is Barry Goddard

2. Mark Hanna says:

The circumference of a circle (i.e. the length of the rope) is related to the radius of that circle (i.e. the distance from the *centre* of the Earth) via the formula C=πd. So, if you increase C by 3 feet, πd also increases by 3 feet. Divide by 2π to convert that to the radius and (because π is about 3) you’re left with about half a foot.

It’s surprising because intuitively one wouldn’t expect the tiny addition of just 3 feet to an overall length of 24,000 miles to result in any noticeable increase in how far off the ground the rope is. The truth, though, is that the original circumference and radius are both irrelevant. If it were 1 inch to start with instead of 24,000 miles the radius would still increase by about 6 inches, just as it would if the initial length of the rope were 1,000 light years.

• Anne Elk says:

Very clever. Thank you for the explanation, Mark.

• mgm75 says:

Thanks for the explanation, quite fascinating when you think about it

• Natasha says:

Thanks Mark. That’s a good explanation, as are Simon’s and Ken’s below.

3. Parax says:

Circumference of a circle is 2 x [Pi] x r
adding 3 ft to the circumference of any circle always adds 3ft / (2xPi) = 5.7inches to the radius.

4. Simon says:

Can anyone explain why and what is surprising about this answer?

The ratio of the circumference of a circle (the length of the rope) to the radius (the distance that the rope is from the Earth’s centre) is 2 × (pi) (or exactly 2 × 22/7 for DAve). This is near enough to 6 for an approximation.

Therefore, if you increase the length of the rope (circumference) by 3 feet, its distance from the Earth’s centre (radius) will increase by approx 1/6 of that, so about 6 inches.

I didn’t find anything surprising about the answer but I imagine that some people would assume that adding 3 feet to 24,000 miles would not make a noticeable difference. Measured relative to its distance from the centre of the Earth, it didn’t make much difference, but when viewed by us tiny things way out on the edge, it does seem like quite a lot.

• Simon says:

Beaten. Thought mine was going to be the first.

• Topher says:

careful with the word “exactly”. 22/7 is just an approximation. http://en.wikipedia.org/wiki/Pi#Continued_fractions

• Simon says:

I’ll try to remember to add a 😉 next time I make a joke.

• Aimée says:

Thank you Simon that’s a very clear explanation.

Aimée

5. jasontimothyjones says:

What Mark said

6. Steve Jones says:

Why does Richard think it’s a surprising result? There’s a linear relationship between the circumference and diameter of a circle. The results exactly what you’d expect.

• Sally Pollard says:

Only if you’re a giant smuggy smartypants.

• Steve Jones says:

Is that your baby-talk term for somebody numerate?

7. Bob Lewis says:

I ended up with 5.99 inches. But I was pretty sloppy with my sig figs throughout on account of doing it on the back of a Taco Bell receipt.

8. Eddie says:

C=2πr
C’=2πr’
C’=2πr+a
2πr’=2πr+a
r’-r=a/2π

9. vldr says:

Metric system, please?

• Evan Stone says:

Ok, so you add 10 feet to the rope. It will then be ~10/6 ft = 1 ft 8 in above the ground. 😉

• Geodetective says:

Imagine the Earth is perfectly circular and has a circumference of 40000 km. If you had a piece of rope stretched around it, touching the floor, that rope would also be 40000 km long. If you were to add 1 meter to that rope and then make it hover above the ground until it’s a perfect circle again, approximately how far off the ground would it get?

The rope would raise 16 cm off the ground if you added 1m to it!

Given the variables:
Initial Circumference (IC), Final Circumference (FC), Added Circumference (AC)
Initial Radius (IR), Final Radius (FR), Added Radius (AR)

We are looking for AR (Added Radius).

IC + AC = FC
FC – IC = AC

IC / pi / 2 = IR
FC / pi / 2 = FR
AC / pi / 2 = AR

AC = 1
AR = 1 / pi / 2 = 0.15915494309189533576888376337251

10. Alma says:

I’m thick at maths so I don’t understand any of this – but it does surprise me that such a comparatively tiny increase would add 6 inches to the distance from the surface. On a very obvious and immediately intuitive level this seems very weird! And now, Richard, please let’s have a few non-maths puzzles for those of us whose brains don’t really work well with this sort of stuff ! Had a lot of maths ones lately – let’s have some nice interesting logic puzzles! 🙂

• Mike Benton says:

The counter-intuitiveness is because you’re not actually comparing like with like. You’re adding 3 feet to 24,000 miles and the effect is ½ foot added to about 4,000 miles (distance to Earth’s centre).

11. Stan says:

My main question is what fraction of respondents hadn’t seen this puzzle before. I suspect it is mentioned in most high school physics classes or wherever they teach c=2 pi r.

12. Christian Geiselmann says:

I got an elevation of 0,477464829 feet, based on the assumption that a mile is 5280 feet and that Pi is 3,14159265359. But nobody really knows what a mile is, right? 😦

• Tim says:

50 miles is what my car tells me I drive every day.

• An Onyx Mouse says:

Weekends too? I want a talking car.

13. Steve Jones says:

You’re problem is the assumed value of Pi. The unit of a mile is a complete irrelevance to the question as the circumference of the Earth is wholly immaterial to the answer which would have been the same whether the object in question have been Jupiter or a dinner plate.

A reasonable case might be made for needing to know how long a foot is in terms of inches. However, even that isn’t strictly necessary as the result could be expressed in factions of a foot.

• Noseache says:

@Jones. It’s “Your” and not “You’re”. It’s “fractions” and not “factions”. Apart from that, your four sentences of aloof self-importance are more or less presentable.

• An Onyx Mouse says:

And “had been Jupiter” instead of “have been Jupiter”.

Maths: 11/10
Speling: 7/10

• Christian Geiselmann says:

Sure, but I wasn’t sure if a mile in the UK is 5280 feet, as the US-driven Google told (metaphorically, Tim!) me when I looked it up.

• Tim says:

@Christian Geiselmann – I see where you are coming from now. Apologies. Too busy chatting with my car this morning.

• An Onyx Mouse says:

I’ve changed my mind, I want a flying car. It’s 2014! – where is my jetcar?

And my rocket pants?

• Stevenz says:

A mile isn’t a mile isn’t a mile. For examples, a nautical mile isn’t 5280 feet, but 6000 feet. When a place kicker misses a field goal “by a mile” how far from the nearest upright was it? How about a country mile? North Michigan Avenue in Chicago is often called the Miracle Mile. Is it a mile long? Yes, Barry is rubbing off on me but he’s the most interesting person here so I don’t mind catching a bit of that.

• Cathy says:

You need to watch the episode of horrible histories where they talk about the London mile and the Oxford mile etc. There used to be so many more miles… I prefer kilometres myself. The name tells you exactly how may metres it has in it.

• Christian Geiselmann says:

Indeed an advantage of kilometres. Funnily, a “mile” (from mille passuum) should mean something similar (1000 paces). But obviously, people who then made up their various “miles” were not that enthusiastic about the strict concept of 1000 = 1000…

14. Steve Jones says:

I know. Unfortunately Richard’s site does not allow post-editing when I spotted the typo. Also, I now see I hit the wrong reply box and the poster was using the convention of a comma for the decimal point.

In any case, his post was sarcastic. It deserves a sarcastic response.

• Noseache says:

Your reply did not read sarcastic; it read self-important. Big difference.

• An Onyx Mouse says:

“I now see I hit the wrong reply box”

Twice! 🙂

• Steve Jones says:

I read it as being (mainly) sarcastic about our culture being wedded to an outdated system of measurement, but never mind. Also you present a false dichotomy; sarcasm is not mutually exclusive with being self-important. Indeed they can be bedfellows. Quite cozy ones.

On the problems, its always interesting to see some of the lateral approaches, but those posts based purely on making points about irrelevances are surely fair game, As, I should add, are those who think it almost admirable to understand what is surely among the simplest of geometric and algebraic principles by throwing around classroom insults.

Nb. I do hate getting punctuation and grammar wrong, not to mention the typos. It comes from a troubled childhood, dreadful physical coordination and being trained to spell phonetically rather than by rote. Unfortunately English is written phonetically. Such mistakes cause me almost physical pain.

• Christian Geiselmann says:

Hello Steve,
It is a little bit off the main topic here, but I would be interested in learning what you mean by “Unfortunately English is written phonetically”? In my understanding, “phonetically written” would mean that one phoneme is represented by one (and ideally only one and the same) grapheme. We find this in languages such as Serbian (etc.) where the written language was constructed, relatively late in history, purposefully in order to match the phonetic repertoire of the spoken language. In English, I would say, things are different, due to the very long development of both written and spoken language in parallel… But perhaps I have mistaken your intention? (This is a serious question, not sarcasm or so… just ot prevent misunderstandings…)

• Noseache says:

@Jones. I take it your post of 7.53 pm was intended for me. I am not even sure where to begin with this latest dollop of nonsense. I have presented no false dichotomy (although fair play to you for actually spelling that word correctly in the alphabet-soup mess of the rest of your post). But to call English a phonetic language is untrue, as thirty seconds of the lightest of light research would have confirmed to you before making this ludicrous claim.

15. Score V says:

Is it just me or should it be spherical and not circular?

• Mike Benton says:

It only needs to be circular around the diameter where the rope is laid so it could be an oblate spheroid or even a torus. Surprised no-one yet has questioned whether “floor” = “ground”.

• Score V says:

But if it’s circular and you make the rope hover above it, the circle formed by the rope does not change in size. If it’s spherical (or the rope moves away from the centre of the circle), then you avoid that problem.

• ashleyfrieze says:

A sphere is a ball. A rope around the earth, would be a ring, or circle.

• Steve Jones says:

Yes, it’s you.

16. ashleyfrieze says:

It’s a lot easier to calculate if you imagine the rope is being extended by approximately PI feet (or metres or whatever). Adding PI to the circumference gives an increase of PI in the diameter, or half-PI in the radius.

The reason it’s surprising is that:

1. It applies to any circle, whether one the size of the earth, or one the size of a coke can.
2. You feel like, in this case, a 6 inch gap all the way around the earth is a lot of space to get from just 3 feet, and that’s down to the fact that our perception of how big the circle is, is related to its area, which grows based on the square of the total of the radius, not just the amount you add on.

17. Ken Haley says:

The surprising thing about the answer is that an insignificant change in the circumference results in an apparently very significant change in the height of the rope. Six inches is quite visible; one might expect it to be microscopic. But that’s a myopic perception. Considering that the original “height” of the earth’s surface is about 4,000 miles from its center, six inches is just as insignificant as the 3-foot change in the 24,000 mile circumference.

Bottom line: because the ratio of circumference to radius is fixed (roughly, 6:1), adding 3 feet to the circumference always adds about 6 inches to the radius — no matter how huge or tiny the circle is.

• Christian Geiselmann says:

This was the most convincing contribution here, from my perspective. Thank you, Ken.

18. Aleksey says:

The answer is actually pretty simple

Here is Math explanation
R – radius of the Earth
C – circumference
pi – is the magical pi number 3.14

2 * pi * R = C

With the new 3 feet, it will look like this:

2 * pi * (R+x) = C + 3
2 * pi * R + 2 * pi * x = C + 3
C + 2 * pi * x = C + 3

x = 3/(2*pi) = 0.4777 feet or 5.73 inches. Roughly 6 inches

The exact measurement is 18/pi inches.

19. laur says:

3 feet/2*3.14

• Bill T. says:

note that parens are required around the 2 pi, because the expression is resolved innermost expression(s) first then left-to-right.

20. L.Long says:

Actually the answer is every value from 0 to 12. Because there is nothing in the question that says it must be exactly centered. It can be touching at one point and maximally distance at the other side.

21. Alma says:

And anyway . . . is a fraction of a foot a toe????

22. What is surprising about the answer (6 inches) is that it depends on pi being 3 rather than 3.1415926etc

• Tim says:

The question asks “…approximately how far off the ground would it get?”. Calculating the exact value using pi is a fairly straightforward maths problem but I wouldn’t call it a puzzle. Using 3 as an approximation of pi and working out the answer mentally is the trick that Richard expects us to work out IMHO. Given that interpretation the “official” answer should be “about 6 inches” not “6 inches”.

• Bill T. says:

@Tim: Der, I missed the “approximately”, so when I got 3 / (2 pi) I thought I missed something.

• Tim says:

Ratio C:d = pi:1 ≈ 3:1
Increasing C by 3 units (furlongs anyone?) extends d by 1 or r by 0.5.
Is where I was coming from…..

23. Liedzeit says:

Surely the measurement system has nothing to do with the problem. (A “problem” by the way I thought everyone had encountered at school.) But can it be a coincidence that the best explanation is by GeoDetective who uses the metric system?

24. GFR says:

I got this wrong because I took the extra 3 feet to be added to the diameter, not the radius. I am still not sure why I am wrong.

• Eddie says:

The rope is around the circumference, not the diameter – or radius, for that matter.

25. GFR says:

Right, but I think of the formula for circumference to be pi*diameter. The right answer is 3 / 2 / Pi. I do not understand why it isn’t 3 / Pi.

• Geodetective says:

That has to do with radius vs. diameter. The extra diameter is 3/pi. And that extra diameter is the total of that on both sides of the moon. Devide by 2, to get one side.
You need the radius, not the diameter.

26. Retired - (but wife says "retarded" says:

Holy cow! I am humbled (and humiliated)! This entire bunch of explanations, replies, and retorts astounds me. You are all so much fun to read. I’m coming here again. Thanks for the fun from beginning to end.

27. Sheldon Cooper says:

In the interest of accuracy, 22/7 as a proxy for Pi, is IMHO totally unacceptable. I suggest that we work with the following approximation (off the top of my head)

pi=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 50244 59455 34690 83026 42522 30825 33446 85035 26193 11881 71010 00313 78387 52886 58753 32083 81420 61717 76691 47303 59825 34904 28755 46873 11595 62863 88235 37875 93751 95778 18577 80532 17122 68066 13001 92787 66111 95909 21642 01989 38095 25720 10654 85863 27886 59361 53381 82796 82303 01952 03530 18529 68995 77362 25994 13891 24972 17752 83479 13151 55748 57242 45415 06959 50829 53311 68617 27855 88907 50983 81754 63746 49393 19255 06040 09277 01671 13900 98488 24012 85836 16035 63707 66010 47101 81942 95559 61989 46767 83744 94482 55379 77472 68471 04047 53464 62080 46684 25906 94912 93313 67702 89891 52104 75216 20569 66024 05803 81501 93511 25338 24300 35587 64024 74964 73263 91419 92726 04269 92279 67823 54781 63600 93417 21641 21992 45863 15030 28618 29745 55706 74983 85054 94588 58692 69956 90927 21079 75093 02955 32116 53449 87202 75596 02364 80665 49911 98818 34797 75356 63698 07426 54252 78625 51818 41757 46728 90977 77279 38000 81647 06001 61452 49192 17321 72147 72350 14144 19735 68548 16136 11573 52552 13347 57418 49468 43852 33239 07394 14333 45477 62416 86251 89835 69485 56209 92192 22184 27255 02542 56887 67179 04946 01653 46680 49886 27232 79178 60857 84383 82796 79766 81454 10095 38837 86360 95068 00642 25125 20511 73929 84896 08412 84886 26945 60424 19652 85022 21066 11863 06744 27862 20391 94945 04712 37137 86960 95636 43719 17287 46776 46575 73962 41389 08658 32645 99581 33904 78027 59009 94657 64078 95126 94683 98352 59570 98258 22620 52248 94077 26719 47826 84826 01476 99090 26401 36394 43745 53050 68203 49625 24517 49399 65143 14298 09190 65925 09372 21696 46151 57098 58387 41059 78859 59772 97549 89301 61753 92846 81382 68683 86894 27741 55991 85592 52459 53959 43104 99725 24680 84598 72736 44695 84865 38367 36222 62609 91246 08051 24388 43904 51244 13654 97627 80797 71569 14359 97700 12961 60894 41694 86855 58484 06353 42207 22258 28488 64815 84560 28506 01684 27394 52267 46767 88952 52138 52254 99546 66727 82398 64565 96116 35488 62305 77456 49803 55936 34568 17432 41125 15076 06947 94510 96596 09402 52288 79710 89314 56691 36867 22874 89405 60101 50330 86179 28680 92087 47609 17824 93858 90097 14909 67598 52613 65549 78189 31297 84821 68299 89487 22658 80485 75640 14270 47755 51323 79641 45152 37462 34364 54285 84447 95265 86782 10511 41354 73573 95231 13427 16610 21359 69536 23144 29524 84937 18711 01457 65403 59027 99344 03742 00731 05785 39062 19838 74478 08478 48968 33214 45713 86875 19435 06430 21845 31910 48481 00537 06146 80674 91927 81911 97939 95206 14196 63428 75444 06437 45123 71819 21799 98391 01591 95618 14675 14269 12397 48940 90718 64942 31961 56794 52080 95146 55022 52316 03881 93014 20937 62137 85595 66389 37787 08303 90697 92077 34672 21825 62599 66150 14215 03068 03844 77345 49202 60541 46659 25201 49744 28507 32518 66600 21324 34088 19071 04863 31734 64965 14539 05796 26856 10055 08106 65879 69981 63574 73638 40525 71459 10289 70641 40110 97120 62804 39039 75951 56771 57700 42033 78699 36007 23055 87631 76359 42187 31251 47120 53292 81918 26186 12586 73215 79198 41484 88291 64470 60957 52706 95722 09175 67116 72291 09816 90915 28017 35067 12748 58322 28718 35209 35396 57251 21083 57915 13698 82091 44421 00675 10334 67110 31412 67111 36990 86585 16398 31501 97016 51511 68517 14376 57618 35155 65088 49099 89859 98238 73455 28331 63550 76479 18535 89322 61854 89632 13293 30898 57064 20467 52590 70915 48141 65498 59461 63718 02709 81994 30992 44889 57571 28289 05923 23326 09729 97120 84433 57326 54893 82391 19325 97463 66730 58360 41428 13883 03203 82490 37589 85243 74417 02913 27656 18093 77344 40307 07469 21120 19130 20330 38019 76211 01100 44929 32151 60842 44485 96376 69838 95228 68478 31235 52658 21314 49576 85726 24334 41893 03968 64262 43410 77322 69780 28073 18915 44110 10446 82325 27162 01052 65227 21116 60396 66557 30925 47110 55785 37634 66820 65310 98965 26918 62056 47693 12570 58635 66201 85581 00729 36065 98764 86117 91045 33488 50346 11365 76867 53249 44166 80396 26579 78771 85560 84552 96541 26654 08530 61434 44318 58676 97514 56614 06800 70023 78776 59134 40171 27494 70420 56223 05389 94561 31407 11270 00407 85473 32699 39081 45466 46458 80797 27082 66830 63432 85878 56983 05235 80893 30657 57406 79545 71637 75254 20211 49557 61581 40025 01262 28594 13021 64715 50979 25923 09907 96547 37612 55176 56751 35751 78296 66454 77917 45011 29961 48903 04639 94713 29621 07340 43751 89573 59614 58901 93897 13111 79042 97828 56475 03203 19869 15140 28708 08599 04801 09412 14722 13179 47647 77262 24142 54854 54033 21571 85306 14228 81375 85043 06332 17518 29798 66223 71721 59160 77166 92547 48738 98665 49494 50114 65406 28433 66393 79003 97692 65672 14638 53067 36096 57120 91807 63832 71664 16274 88880 07869 25602 90228 47210 40317 21186 08204 19000 42296 61711 96377 92133 75751 14959 50156 60496 31862 94726 54736 42523 08177 03675 15906 73502 35072 83540 56704 03867 43513 62222 47715 89150 49530 98444 89333 09634 08780 76932 59939 78054 19341 44737 74418 42631 29860 80998 88687 41326 04721 56951 62396 58645 73021 63159 81931 95167 35381 29741 67729 47867 24229 24654 36680 09806 76928 23828 06899 64004 82435 40370 14163 14965 89794 09243 23789 69070 69779 42236 25082 21688 95738 37986 23001 59377 64716 51228 93578 60158 81617 55782 97352 33446 04281 51262 72037 34314 65319 77774 16031 99066 55418 76397 92933 44195 21541 34189 94854 44734 56738 31624 99341 91318 14809 27777 10386 38773 43177 20754 56545 32207 77092 12019 05166 09628 04909 26360 19759 88281 61332 31666 36528 61932 66863 36062 73567 63035 44776 28035 04507 77235 54710 58595 48702 79081 43562 40145 17180 62464 36267 94561 27531 81340 78330 33625 42327 83944 97538 24372 05835 31147 71199 26063 81334 67768 79695 97030 98339 13077 10987 04085 91337 46414 42822 77263 46594 70474 58784 77872 01927 71528 07317 67907 70715 72134 44730 60570 07334 92436 93113 83504 93163 12840 42512 19256 51798 06941 13528 01314 70130 47816 43788 51852 90928 54520 11658 39341 96562 13491 43415 95625 86586 55705 52690 49652 09858 03385 07224 26482 93972 85847 83163 05777 75606 88876 44624 82468 57926 03953 52773 48030 48029 00587 60758 25104 74709 16439 61362 67604 49256 27420 42083 20856 61190 62545 43372 13153 59584 50687 72460 29016 18766 79524 06163 42522 57719 54291 62991 93064 55377 99140 37340 43287 52628 88963 99587 94757 29174 64263 57455 25407 90914 51357 11136 94109 11939 32519 10760 20825 20261 87985 31887 70584 29725 91677 81314 96990 09019 21169 71737 27847 68472 68608 49003 37702 42429 16513 00500 51683 23364 35038 95170 29893 92233 45172 20138 12806 96501 17844 08745 19601 21228 59937 16231 30171 14448 46409 03890 64495 44400 61986 90754 85160 26327 50529 83491 87407 86680 88183 38510 22833 45085 04860 82503 93021 33219 71551 84306 35455 00766 82829 49304 13776 55279 39751 75461 39539 84683 39363 83047 46119 96653 85815 38420 56853 38621 86725 23340 28308 71123 28278 92125 07712 62946 32295 63989 89893 58211 67456 27010 21835 64622 01349 67151 88190 97303 81198 00497 34072 39610 36854 06643 19395 09790 19069 96395 52453 00545 05806 85501 95673 02292 19139 33918 56803 44903 98205 95510 02263 53536 19204 19947 45538 59381 02343 95544 95977 83779 02374 21617 27111 72364 34354 39478 22181 85286 24085 14006 66044 33258 88569 86705 43154 70696 57474 58550 33232 33421 07301 54594 05165 53790 68662 73337 99585 11562 57843 22988 27372 31989 87571 41595 78111 96358 33005 94087 30681 21602 87649 62867 44604 77464 91599 50549 73742 56269 01049 03778 19868 35938 14657 41268 04925 64879 85561 45372 34786 73303 90468 83834 36346 55379 49864 19270 56387 29317 48723 32083 76011 23029 91136 79386 27089 43879 93620 16295 15413 37142 48928 30722 01269 01475 46684 76535 76164 77379 46752 00490 75715 55278 19653 62132 39264 06160 13635 81559 07422 02020 31872 77605 27721 90055 61484 25551 87925 30343 51398 44253 22341 57623 36106 42506 39049 75008 65627 10953 59194 65897 51413 10348 22769 30624 74353 63256 91607 81547 81811 52843 66795 70611 08615 33150 44521 27473 92454 49454 23682 88606 13408 41486 37767 00961 20715 12491 40430 27253 86076 48236 34143 34623 51897 57664 52164 13767 96903 14950 19108 57598 44239 19862 91642 19399 49072 36234 64684 41173 94032 65918 40443 78051 33389 45257 42399 50829 65912 28508 55582 15725 03107 12570 12668 30240 29295 25220 11872 67675 62204 15420 51618 41634 84756 51699 98116 14101 00299 60783 86909 29160 30288 40026 91041 40792 88621 50784 24516 70908 70006 99282 12066 04183 71806 53556 72525 32567 53286 12910 42487 76182 58297 65157 95984 70356 22262 93486 00341 58722 98053 49896 50226 29174 87882 02734 20922 22453 39856 26476 69149 05562 84250 39127 57710 28402 79980 66365 82548 89264 88025 45661 01729 67026 64076 55904 29099 45681 50652 65305 37182 94127 03369 31378 51786 09040 70866 71149 65583 43434 76933 85781 71138 64558 73678 12301 45876 87126 60348 91390 95620 09939 36103 10291 61615 28813 84379 09904 23174 73363 94804 57593 14931 40529 76347 57481 19356 70911 01377 51721 00803 15590 24853 09066 92037 67192 20332 29094 33467 68514 22144 77379 39375 17034 43661 99104 03375 11173 54719 18550 46449 02636 55128 16228 82446 25759 16333 03910 72253 83742 18214 08835 08657 39177 15096 82887 47826 56995 99574 49066 17583 44137 52239 70968 34080 05355 98491 75417 38188 39994 46974 86762 65516 58276 58483 58845 31427 75687 90029 09517 02835 29716 34456 21296 40435 23117 60066 51012 41200 65975 58512 76178 58382 92041 97484 42360 80071 93045 76189 32349 22927 96501 98751 87212 72675 07981 25547 09589 04556 35792 12210 33346 69749 92356 30254 94780 24901 14195 21238 28153 09114 07907 38602 51522 74299 58180 72471 62591 66854 51333 12394 80494 70791 19153 26734 30282 44186 04142 63639 54800 04480 02670 49624 82017 92896 47669 75831 83271 31425 17029 69234 88962 76684 40323 26092 75249 60357 99646 92565 04936 81836 09003 23809 29345 95889 70695 36534 94060 34021 66544 37558 90045 63288 22505 45255 64056 44824 65151 87547 11962 18443 96582 53375 43885 69094 11303 15095 26179 37800 29741 20766 51479 39425 90298 96959 46995 56576 12186 56196 73378 62362 56125 21632 08628 69222 10327 48892 18654 36480 22967 80705 76561 51446 32046 92790 68212 07388 37781 42335 62823 60896 32080 68222 46801 22482 61177 18589 63814 09183 90367 36722 20888 32151 37556 00372 79839 40041 52970 02878 30766 70944 47456 01345 56417 25437 09069 79396 12257 14298 94671 54357 84687 88614 44581 23145 93571 98492 25284 71605 04922 12424 70141 21478 05734 55105 00801 90869 96033 02763 47870 81081 75450 11930 71412 23390 86639 38339 52942 57869 05076 43100 63835 19834 38934 15961 31854 34754 64955 69781 03829 30971 64651 43840 70070 73604 11237 35998 43452 25161 05070 27056 23526 60127 64848 30840 76118 30130 52793 20542 74628 65403 60367 45328 65105 70658 74882 25698 15793 67897 66974 22057 50596 83440 86973 50201 41020 67235 85020 07245 22563 26513 41055 92401 90274 21624 84391 40359 98953 53945 90944 07046 91209 14093 87001 26456 00162 37428 80210 92764 57931 06579 22955 24988 72758 46101 26483 69998 92256 95968 81592 05600 10165 52563 7567

• An Onyx Mouse says:

You misplaced one of the 7’s.

• Tim says:

That should be close enough for practical purposes 🙂

28. Christian Geiselmann says:

It is never late to contribute some exactitude to a scholarly discussion.

29. Christian Geiselmann says:

PS. I think this prooves that there is no God: If God existed, would he or she have created circles that have such a ridicolous ratio of diameter and circumference? Hard to imagine…

• Ken Haley says:

Au contraire: What if God has left us a secret message in the decimal expansion? Suppose, say, after trillions of trillions of digits in the expansion, there’s a sequence of all 1’s and 0’s (and no other digit) that results in a message spelled out when the digits are arranged in a 1000×1000 grid? That would be irrefutable evidence that God exists.

That’s fantasy; but seriously, I think it would be impossible for God to have pi be any other value that what it is. It would be just as impossible as making 2 + 2 equal anything besides 4. It’s the very nature of the numbers–not something God did or did not do.

• Christian Geiselmann says:

If!

• ctj says:

Ken, not only does every possible “message” gets spelled out in a sequence of 0s and 1s when arranged in a 1000×1000 grid, every such message happens an infinite number of times.

that’s because pi is irrational.

30. Christian Geiselmann says:

As for the question if God Almighty could or could not create an universe where pi is exactly 3 (or anything else more tidy than the odd pi of our universe), I would say: shouldn’t be impossible even if you think in terms of physics: multidimensional space, curved dimensions, etc. gadgeds of string theory and the like. But if there is a God, he, she or it should be almigthy, and if h/s/i is almighty, what would hinder h/s/i to create such an universe?

Of course, again: if !.

31. Christian Geiselmann says:

Uh… here is a solution to the God-and-his-odd-Pi problem:

God’s first universe of course had Pi = 3. But then came the fall of man, and ever since we are living in a basically disarranged universe, with even a Pi that never ends. So, if you like, you may take the deplorably lenghty Pi as a proove that God exists… 🙂

32. Christian Geiselmann says:

Last contribution to this topic for today: Ken, I am afraid, even if you find in the long, long array of numbers that are behind the comma in Pi a section with 1 million consecutive 1s and 0s which form, arranged in a square 1000 x 1000, a message that you can read in your preferred language (by the way: which character encoding would be used?), this would prove nothing: under the assumption that Pi does never end, there MUST be, anywhere, far, far away in the queue of Pi a section where such a consecutive array of 0s and 1s occurs. Given the infinite length of the string…

I even feel a little bid sad about the loss of a very nice proof for God’s existence… (not that I bothered, but I liked the argument.)

• ChrisR says:

quote […there MUST be, anywhere, far, far away in the queue of Pi a section where such a consecutive array of 0s and 1s occurs. Given the infinite length of the string…]

what is surprising is that, if the expansion of pi really is infinite, such a consecutive array of 0s and 1s will occur an infinite number of times !

• Christian Geiselmann says:

Um… or could God possibly have outruled such occurences? Then, THIS would be another proof of God’s existence 🙂

• The Anti-Pi says:

I think pi’s overrated.

Discuss.

“I am the Anti-Pi
I am an anarchist
Don’t know what I want
But I know how to get it
I wanna destroy passerby”

• An Onyx Mouse says:

Which god are you referring to? There are so many.

• Ken Haley says:

Well, maybe not the last..:)

It’s not true that such any sequence exists, just because the decimal expansion is infinite, even with the constraint that the expansion be non-repeating. Consider the following number with a non-repeating, infinite decimal expansion thus:
.202002000200002000002…
where the number of zeros increases by one each time. Nowhere will you find ANY sequence of 1’s and 0’s, let alone the one we’re imagining.

It’s a common misconception that an infinite set of things must contain all possible things that COULD belong to the set. The set of real numbers between 0 and 1 is infinite, but that set does not contain the real number, 2. Similarly, just because the decimal expansion of pi is infinite, it doesn’t PROVE that any given finite sequence must exist therein.

Now, however, having said that, I admit that it is highly likely that any finite sequence WILL exist somewhere in pi’s decimal expansion, assuming there’s no unknown bias in the nature of pi that would prevent it.

• Christian Geiselmann says:

Good argument, well presented.

• Mike Benton says:

Ooh, I missed this. Yes, Ken is definitely correct when he says that the decimal (or any other base) expansion of pi doesn’t necessarily contain every possible sequence of digits. It’s an unanswered question although the generally accepted answer is “probably”.

It’s actually a mistake to refer to pi’s digits as “random” since pi has a fixed value and is not truly random. The sequence of digits are unpredictable but they are not random, although they do have the appearance of randomness (as far as they have been checked). However, we don’t even know if the digit ‘1’ appears exactly one tenth of the time in the infinite series or whether it appears an infinite number of times, both of which would be the case if the sequence were truly random.

With regard to an embedded secret message being proof of an intelligent creator, if the digits of pi do truly contain every possible sequence of digits, then they will contain an infinite number of messages, messages that contradict each other, messages that are self-contradictory, messages that are verifiably false, and those same messages with typos, spelling mistakes, poor grammar… hang on a minute – I think the World Wide Web proves the existence of an intelligent creator!

• Christian Geiselmann says:

Well… then… couldn’t it be that the intelligent creator had been come into existence at random?

• Starman says:

Messages that contradict each other or are false, with typos, spelling mistakes, and poor grammar? Sounds more like Goddard than God.

• GrumpMeister Flash says:

This has been a fascinating discussion. Thanks for posting.

33. G. Almighty says:

Permit me to say a few words here ………

34. Eddie says:

22/7 differs from π by about 0.04%.
355/113 differs from π by only about 0.000008%.

35. Eddie says:

Good puzzle this – maddening but great when you solve it.

36. Pat Harkin says:

Barry:

Your intuitive answer works for a single additional loop. It also works for two loops half as long positioned 180 degrees apart. But it doesn’t work for three loops one third the size 120 degrees apart or any higher number (though it does work for n loops all in a line). That would suggest your “intuitive” answer is a coincidence, rather than a provable, generalisable, mathematical answer.

37. ctj says:

july 25 puzzle: “10 bets you will almost win”? doesn’t that mean you will lose?

38. Ken Haley says:

Typo in my last post: “…such any sequence…” should read “…any such sequence…”

39. bag of cows says:

Is it me…… or are none of you bothered that a rope would be hovering above the earths surface with no visable means of support? ……… would health and safety have to put a ‘caution trip hazard’ sign every ten feet along its length? and how many signs would it take to stop someone tripping over and putting a claim in?

• Starman says:

There is no number of signs that would stop every idiot from tripping over the rope.

• Christian Geiselmann says:

I have a problem with measuring the length of that rope (or its elevation above ground, respectively). Supposed, as kosmologists tell us, that the universe expands continuously, and with it all things in it, obviously Earth’s diameter (or circumference, respectively) constantly grows; so, how can we then possibly define the length of such a rope at all?

• Mike Benton says:

The expansion of the universe (actually “space”) doesn’t mean that all things in it also expand. That is how we actually know that space is expanding – because the distances between things (generally galaxies) is greater now than it was in the past. If absolutely everything expanded, the distances would appear to be the same

To clarify – if all things expanded, this would include all of our rulers, measuring tapes, micrometers, etc, and our sense of how long anything was, so that it would be impossible to detect any change.

• Christian Geiselmann says:

So the suggestion is: space expands, but things in it do not?
But wouldn’t this mean that space and things are separate?
I rather would believe space is everywhere, also “in” things (so to say), and if space expands, things should be part of that process. No?

• Anonymous says:

How would quantum mechanics still work if everything was expanding? What about Planck’s constant and the speed of light?

40. M says:

Metric system, anyone?

41. Christian Geiselmann says:

Well… perhaps my supposition was wrong: Does the universe expand, but things in it remain the same size?

42. Taylor Wright says:

it seems that this answer is correct if Pi is considered to equal 3 (rounded down)