## Correction of the year

…and maybe of all time, at least in quantitative terms. In the New York Times Magazine, 4/10/2016:

An article on March 20 about wave piloting in the Marshall Islands misstated the number of possible paths that could be navigated without instruments among the 34 islands and atolls of the Marshall Islands. It is 561, not a trillion trillion.

A trillion trillion is presumably 1024, and 561 is 5.61 x 102, so the original number was off by a factor of about 1.78 x 1022, which is more than a thousand times greater than the estimated number of grains of sand in all the beaches and deserts of planet earth.

The obligatory screen shot:

I don't know how either the original number or the revised number were calculated. One hypothesis might start with 34!, which is about 2.952328e+38, but that seems too big even for the version of trillion that makes it 1018 rather than 1012. And (34^2)/2 is 578, not 561. Ideas?

[h/t John Burke]

1. ### Dick Margulis said,

April 10, 2016 @ 12:07 pm

34 × 33 / 2 = 561

[(myl) Of course. 34^2 would assume that it's a noteworthy accomplishment to navigate between a given island and itself. Which would probably be true for some of us…]

2. ### Daniele A. Gewurz said,

April 10, 2016 @ 12:28 pm

33! ≈ 8.7 × 10^36, so it just might be (in the “ancient” trillion use) that someone chose a fixed island as a starting point, and then considered all the possibile ordering of the remaining ones as possible paths. Or, someone read 561 as 5^61 (≈ 4,3 × 10^42) which, though far exceeding 10^24 or 10^36, could perhaps colloquially be described as “a trillion trillion” (I for one know that more than once the opposite happened to me; writing for instance that Avogadro number is something times 10^23, I then found in the galleys or the printed version that it was something times 1023).
As for 561 itself, it is perhaps just an empirical number found by investigating the actual position of the islands?

3. ### Arthur David Olson said,

April 10, 2016 @ 12:29 pm

34 x (33 /2) (expressed differently, and perhaps more intuitively)
33+32+31+…+3+2+1 = 561

4. ### Rebecca said,

April 10, 2016 @ 12:37 pm

Maybe original writer wrote a draft in which he didn't yet know the number, so he inserted an obviously ridiculously incorrect place holder to make sure it didn't get out without editing. Except it did.

5. ### Guy said,

April 10, 2016 @ 12:40 pm

As Gewurz says, 10^36 is a trillion trillion on the "long scale" interpretation of trillion. So I think this is just a matter of dueling interpretations of "paths". I do find the "corrected" claim – that no human could possibly memorize 561 paths – kind of implausible, though. I think that should totally be within the capabilities of an ordinary person who frequently travels them.

[(myl) It does make sense that the "long scale" trillion is what the writers originally had in mind, which would mean that they were off by a factor in the general range of the number of atoms in the earth. As for memorizing the pairwise routes, that could be easier than (34*33)/2, anyhow, since the way to get from A to C may often be the composition of A to B and B to C…]

6. ### David Fried said,

April 10, 2016 @ 1:21 pm

I read the original article and was struck by the unlikelihood of "trillions" of paths. The article was about ancient Polynesian methods of navigating under sail by, among other things, judging the set and power of the currents as they are reflected from the various islands, atolls and reefs. This is not an exercise in pure geometry–i.e., there are not 561 or 578 routes. Rather the available routes are closely constrained by wind and current. It is likely that on any given day it is prohibitively time-consuming and difficult to get from one particular island to another within an archipelago like the Marshalls. So there is no question of memorizing 500+ sea routes, and the navigator, unlike Western sailors, doesn't aspire to know exactly where he is at all times. Rather, his ability to read wind and wave enables him to infer the position of islands well over the horizon. In effect, he's shooting for a target much larger than that available using Western celestial/trigonometric methods. So the ancient Polynesians could make voyages of exploration seeking entirely unknown island chains and still be confident of returning home.

7. ### Jamie said,

April 10, 2016 @ 3:20 pm

"BBC More or Less: How many different tweets can be created from 140 characters?"
http://www.bbc.co.uk/programmes/p03fm36p

The original estimate was off by about 10^400 !

8. ### John Roch said,

April 10, 2016 @ 3:24 pm

This is elementary permutations and combinations stuff.

As shown above:
Start from any one of 34 islands; go to any of the other 33; can be done in either direction. Hence

34*33/2 = 561

ie

(34! / 32!) / 2

9. ### KWillets said,

April 10, 2016 @ 3:30 pm

This blog may be the source: http://artofwayfinding.blogspot.com/2015/07/the-tradition-of-wave-piloting-marshall.html

There are 24 coral atolls in the Marshalls. If each atoll connects to the remaining 23 atolls, that island alone has to have 46 pairs of opposing currents, if we take the concept literally. This implies something like Avagodro's number of (6.02 x 10^23) of different opposing swells. But, the navigators tell us that there are only 4 significant swells that they consider. How can this be?

10. ### Sili said,

April 10, 2016 @ 4:29 pm

more than a thousand times greater than the estimated number of grains of sand in all the beaches and deserts of planet earth.

Or the number of molecules in two tablespoons of water.

April 10, 2016 @ 9:15 pm

Why shouldn't the "correct" number be 1122 rather than 561? Surely going from Island A to Island B and going from Island B to Island A are different navigational tasks?

12. ### Jason said,

April 10, 2016 @ 9:48 pm

The first is all paths, and the second is all shortest paths. Discrepancy solved!

13. ### RQA said,

April 11, 2016 @ 10:23 am

Rebecca is correct. "Trillion trillion" is a placeholder meant to be edited out. See Otto Friedrich, "There Are 00 Trees in Russia," Harper's Magazine (October 1964).

14. ### DWalker said,

April 11, 2016 @ 3:01 pm

What does "without instruments" have to do with the number of ways those islands can be navigated? I'm truly mystified by this. someone please enlighten me.

15. ### Ken said,

April 13, 2016 @ 1:51 pm

@DWalker: "Without instruments" is different grammar puzzle. It's a phrase that doesn't modify the sentence at all, since the number of paths is the same with or without instruments. Is there a grammatical term for this – maybe just "unnecessary"?

16. ### Rodger C said,

April 14, 2016 @ 6:54 am

@DWalker: I've never sailed in my life, but since no one else has answered you, I think it's that sailing without instruments is a more constrained process than looking at your GPS.

17. ### Andrew (not the same one) said,

April 14, 2016 @ 4:47 pm

Yes, but in that case the number of paths which can be navigated without instruments cannot be calculated from the number of islands.

18. ### andyb said,

April 16, 2016 @ 3:17 pm

33! being close to a long trillion trillion makes some sense, except… why would a writer for the NYT be using long trillions? In my experience, while English people may get confused between long and short counts, most Americans don't even realize there is such a thing as the long count.

Also, while 8.7 trillion trillion is within an order of magnitude of a trillion trillion, people who aren't used to thinking on those scales usually aren't comfortable throwing out a factor as big as 8, are they?