Ever have a challenging pastime to which you applied analytical skills and ended up making it so easy it ceased to be fun?  This happens to me occasionally.  A couple years back we had a sort of "puzzle craze" in the United Stated over "sudoku", a sort of numeric logic puzzle originating from Japan.

The basic puzzle is a 9x9 box of squares in which some of the squares have been filled in with numbers.  Your assignment is to fill in the rest of the squares by following 3 simple rules: (1) each number can appear only once in each row, (2) each number can appear only once in each column, (3) each number can appear only once in each "major square".  The major squares are nine 3x3 boxes superimposed over the 9x9 puzzle in a non-overlapping fashion (so the first major square is formed by columns 1, 2, and 3 where they pass through rows 1, 2, and 3, the next at the intersection of columns 4, 5, and 6 with rows 1, 2, and 3, and so on.)  Since every row contains 9 squares, and every column contains 9 squares, and every major square contains 9 squares, the only numbers you will ever add to the puzzle are the numbers 1 to 9.

There are larger and more complex variants on the basic sudoku, but the above is the most popular and familiar variety.  I did sudokus for awhile but eventually tired of them and moved on to new pastimes.

One reason I tired of them is that a large part of sudoku is simple, repetitive, busy work.  You eliminate the possibilities for each square until a solution pops out for one square and then you fill that in and go back and reevaluate the possibilities for the other squares.  For example, a given square can be any number between 1 and 9.  But if you look at the column that square is in and find 1, 5, and 6 are already used, those are no longer possibilities for that square.  If the row that square is in contains 2, 3, and 7, then those possibilities are also eliminated.  Further if the major square it is in already contains 4 and 9, then those possibilities are eliminated too, leaving only 8, so in this case the square must be 8.

It's much more interesting when you start getting into the more complicated bits of reasoning that involve multiple squares.  Like for instance if a major square is filled in except for two squares, then those two squares have only two possibilities.  Let's say the possibilities are 5 and 9.  Further assume that these two squares are in the same row (or column).  Immediately we can say of all the other squares in that row (or column) that none can be 5 or 9 since those two options have to be used up by those two squares.  But getting to that point involves getting through a period of tedious repetition.

But when you are stuck in a cabin in the rain for a day, away from the amenities you typically would enjoy, even a tedious pastime is welcome.  However, I've found one thing consistently true among software engineers: they generally hate doing tasks by hand which would be better done by a simple computer program.  So I was plonking along with sudoku puzzles, eliminating options by hand when I said "this is stupid, I could make a simple spreadsheet macro to do this for me."

So I built a spreadsheet that would show me the remaining possibilities for each row and column as a row or column header.  Then when I was eliminating possibilities for major squares I said "this is stupid, I can use the same macro to do this for me."  So I added a bit to my spreadsheet to show me the possibilities remaining for every major square.  So now for a given cell I could compare the possibilities for its row, to the possibilities for its column, and the possibilities for its major square and the intersection of those three would represent the possibilities for the cell.  (For example, if the cell is in a row with 4,5,6,7 available, and a column with 2,3,5,7,8 available, and a major square with 1,5,7,9 available, the possibilities for that cell are only 5 and 7, because that is the intersection of the three sets of possibilities.)

I suppose you can guess what happened next.  "This is stupid, I can write a simple function to take the intersection of these possibilities."

One other basic sudoku-solving technique is to look for "loners".  A loner is the only square in a row (or column or major square) that includes a particular number as one of its possibilities.  For example if you have four unfilled squares in a major square and their possibilites are (2,3,4), (2,3,5), (3,5), and (2,5) you can say the first one has to be 4.  Even though 2 and 3 are possibilities for this square, since it is the only square that can be 4, then it must be 4. Finding loners is also tedious, and you can imagine what I did as a result. "This is stupid, yadda yadda yadda..."

So now I had a spreadsheet that would eliminate all the impossible values for a square based on simple cross checking and which would identify loners for me.  But I found as I was entering puzzles that this solved so much of the puzzle automatically, there was very little left for me to do.  The spreadsheet would notify me when I caused an error (forced the possibilities for a cell to dwindle to nothing) and in those cases where I needed to make a guess to explore one possible solution, the spreadsheet would allow me to snapshot the puzzle at that point so I could get back to it if the puzzle didn't work out.  With these features available puzzles rated "difficult" were taking under 2 minutes to solve.  After 5 or 6 of them I shut Excel done and was done with sudoku for the day.  Why bother?  It was too easy now.

The pastime had become writing the code to perform the tedious operations.  Once that was accomplished I was no longer interested in performing those operations.  Go figure.

This is an interesting thought experiment and it involves statistics and some figuring, so I welcome any of my smart friends or anybody who cares to think about it to chime in here.

Assume that sentient civilizations with the ability to transmit some form of radiant communications signals (radio, light, whatever) started appearing in our galaxy as early as 4.5 billion years ago (when the Earth was forming), and assume that they continue to appear (each on its own evolutionary trajectory) for another 4.5 billion years.  Further assume that over this timespan of 9 billion years a total of one million different intelligent civilizations begin transmitting radiant communications. After a random time period of 100 to 1000 years each civilization stops transmitting radiant signals, either because they move on to a better technology that does not require radiant communication (pretty much necessary for interstellar communications) or the civilization dies out.  Finally given that our galaxy's diameter is about 100,000 light years, assume that any of these civilizations when transmitting would be at some random distance from us which does not exceed 100,000 light years (yes I know that's a little big since we aren't at the very edge of the galaxy, but bear with me.)

Now, here we are, smack-dab in the middle of that timescale, halfway between year 1 and year 9 billion.  We turn an array of radio telescopes to the sky and listen in all directions simultaneously, constantly, for a period of 1,000 years.

Is our sky silent?  Or is it noisy?  With a million civilizations out there at some point or another, all sending signals out at different times, what are the chances we would hear none of them at all even if we listen for 1000 years?

I threw together a crude simulation last night to try and answer this question, and I have a result which you might find surprising, but I'm curious to know what other people think first.  Care to venture a guess?  Or perhaps create your own sim and see what results you get?  I'll post my results tomorrow.

So early this morning I couldn't sleep and I ended up writing an article on flickr in one of the many "what lens should I buy" discussions that goes on there.  I have noticed, in my days on flickr, that certain pieces of equipment and certain techniques have a following, and often get recommended simply because of the following rather than because the equipment/technique is actually suited to the purpose of the person asking.  In an effort to supply a counterbalancing opinion, I found myself in need of trigonometry.

The Argument

One such "cult" item is the "nifty fifty" (the EF 50mm f/1.8) lens made for Canon EOS cameras.  It is very sharp, very fast, and very cheap ($80).  If you are on a tight budget (or even if you aren't) it makes sense to have one for your EOS camera unless you have a better 50mm prime, or don't need a 50mm prime.

While I will not argue that it is probably one of the best value-for-money lenses, it is not versatile at all, and yet it seems to get hailed as a magic-bullet lens.  I regularly see people making claims like "it never comes off my camera".  And after having used it myself, I can only conclude that these people shoot one type of thing and one type of thing only, or it never comes off because they don't own any other lenses.

There is a certain love affair with the 50mm focal length because it was the standard focal length for 35mm film for decades.  But in the age of digital SLR cameras, things are different for the less expensive consumer DSLRs.  These DSLR's tend to use an image sensor that is smaller than 35mm film.  The APS-C style sensor, or crop sensor, does not render the entire image cast by a standard lens, but only a smaller piece in the center.  This results in an apparent magnification factor of 1.6.  Hence if you put a 50mm lens on a crop-sensor camera, it's like working with an 80mm lens (50 x 1.6 = 80).  The end result is a smaller-than-expected "field of view" (FOV).

On an old Canon 35mm film camera, a 50mm lens has a FOV of 46°.  But on a crop-sensor camera the FOV is a hair under 29°.  This loss of over a third of the FOV means that on crop-sensor cameras the EF 50mm f/1.8 lens has distinct limitations as to how much you can fit in the frame.

Fortunately, on my EOS 5D the 50mm behaves as expected. Because the 5D is a full frame camera, its sensor is the same size as a 35mm film frame.  So I get 46° out of my EF 50mm, just as nature intended.

The "nifty fifty" on crop-sensor cameras is often described as a "portrait lens".  With the crop factor, the 50mm lens behaves like an 80mm lens, and 80mm is ideal for portraits.  But if you want to shoot anything larger than a head-and-shoulders portrait with the EF 50 1.8 on your Rebel XT or 30D, you'd better have a lot of room behind you, because you are going to need to back up... a lot.

But how much?

The Trigonometry

Well that's where the trig comes in (you can skip this section if you don't want to see how I figured it out).  In order for me to say how much, I needed to be able to reliably compute the distance necessary to view an object of a given width.  But how?  I started by drawing a diagram like this one:

V is my viewing angle.  Okay it's not 29° (or 28.98333° which is the actual FOV of the nifty fifty on a crop sensor), but close enough.  The legs of the triangle extending out from V represent the edges of my FOV as the distance to the subject (marked by the dashed line, d) grows.  The base of the triangle (marked as w) is the width of the field of view at the distance d.  Basically this is a representation of the wedge or cone of that falls within a particular FOV, in this case 30°.

I can pick any distance I want for d, but what I really need is a way to say what d should be to accomodate a subject of a certain width.  In other words, to fit a subject 10 feet wide in my viewfinder, how far back do I need to stand with the nifty fifty on my EOS 30D camera? I supposed that given a formula for that, I could solve the formula for the width so that one could also compute the maximum width viewable given a distance.

The triangle depicted above is an isoceles triangle, as both the legs are the same length, and consequently the angles where the legs meet the base is also to the same.  I spent some time looking online for computations for isoceles triangles, but what I was looking for didn't appear (namely, given the length of the base, and the angle of the peak, what is the height or altitude of an isoceles triangle?)

I studied trig over 20 years ago so I remember very little of it, but I did remember there were a lot of simple equivalences for right triangles (that is, triangles where one of the angles is 90°).  And I realized while looking at my diagram that the line I had drawn to represent the distance, bisected V and split the triangle into 2 right triangles, each of which looked like this:

Bisecting V gives me a 15° angle (V'), and a base width exactly half of what it was before (w').  So if I could take a given distance d and come up with a formula for w', then I should be able to solve that formula for either d or w', keeping in mind that V' is V/2 and w' is w/2.

Doing a quick check online I found the two rudimentary trignometric equivalences for right triangles: for either of the angles other than the 90° one, the sin of that angle is equal to the length of the opposite side divided by the length of the hypoteneuse, and the cos of that angle is equal to the length of the adjacent side divided by the length of the hypoteneuse.  Here are those equivalences for the right triangle above:

Sin V' and cos V' I can get with a pocket calculator, and I'm going to pick a value for either d or w' and solve for the other.  I can solve the equation on the left for w' [w' = (sin V') * h] and I can solve the equation on the right for d [d = (cos V') * h], but both of these solutions require me to know what the hypoteneuse of this triangle is.

But in order to get w' from d or d from w' I need to do more work, mostly because I am not going to know what the hypoteneuse is.  I'm only going to be starting with either V' and w' or V' and d.  So what I need to do is solve one of the equations for h, and then plug that into the other equation.  That should give me a formula I that I can use to solve for either d in terms of w' and V' or w' in terms of d and V'.  So I picked the equation on the right.  Solving that for h gives h = d / (cos V').

So I should be able to substitute d / (cos V') in the equation on the left, like so:

Now I'm good.  I know what V' is, I can get sin V' or cos V' from my calculator, and I am going to pick either d or w'.  So now I can solve for either one, like so:

Done, right?  Well, yes, if I want to know what the appropriate distance is for half the width of my subject using a lens with half the field of view.  Now I want to substitute in the equivalences that w' = w/2 and V' = V/2.  In the equation on the right that will put w/2 on the left of the equal sign, so I will multiply both sides by 2 to solve the equation for w.  That gives me:

Okay they probably aren't the cleanest formulas in the world, but they work and let you get the height of an isoceles triangle from its base width and peak angle, or vice versa.  Using these formulas I could handily compute the needed distance for a given width in a given field of view, and this allowed me to present something more concrete than "gee whiz, that EF 50mm 1.8 is awfully confining on a crop sensor camera."

Back to the Argument

So how confining is that nifty fifty?

5 feet wide = 9.7 feet away
10 feet wide = 19.3 feet away
15 feet wide = 29 feet away
20 feet wide = 38.7 feet away
25 feet wide = 48.4 feet away

Pretty confining!  If you are trying to capture 3 people sitting on a couch which is 8 feet long all in one shot, you need to stand 15 feet 6 inches away.  Better have a big living room, or one where there isn't a TV 10 feet from the couch.  Or maybe if you moved the couch outside... that would be cool for an album cover, but for Aunt Bea, Uncle Joe, and Granny, it is probably less so.

Working with the EF 50mm f/1.8 is a good exercise though for learning how to push a lens to do what you need, and it's plain old good exercise, because you're going to be backing up a lot.  You can get that 8 foot couch in shot if you shoot from an angle, but then you will need to stop your aperture down to widen up the depth of field so that everyone will be in focus... which means you can't shoot low light anymore so you might need lamps or a flash.  Or you could give up on that shot and shoot the people individually.

Or, you could simply not get the EF 50mm 1.8 in the first place, if you are not planning to shoot primarily portraits.  If you want to consider the traditional FOV that the great 35mm film artists shot with, you need a lens that gives a FOV on a crop-sensor camera similar to a 50mm lens on a 35mm film camera (i.e. 46°).  The closest bet would be a 28mm lens, like the EF 28mm f/1.8 or EF 28mm f/2.8.  These have a crop-sensor FOV of 47.25°.  With one of these lenses the width to distance figures look like this:

5 feet wide = 5.7 feet away
10 feet wide = 11.4 feet away
15 feet wide = 17.1 feet away
20 feet wide = 22.8 feet away
25 feet wide = 28.6 feet away

Much more reasonable.  And quite interesting how the distance to subject is almost the same as the width of the subject.  No surpise that the 50mm lens became the standard on the cameras of old.

Definition of "undercutting":

Me:

http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=120163294055
Starting time: Sep-27-07 19:45:00 PDT (scheduled, so as to begin after other auctions ended)
Starting bid: US $49.99
Duration: 7-day listing

The other guy:

http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=170153862318
Starting time: Sep-27-07 20:27:48 PDT
Starting bid: US $0.89
Duration: 7-day listing

I've been so busy with work and trying to sell cards that I haven't had time to do much photography.  I did get some great pix in Hyannis a couple weeks ago and some more good stuff at a reception for a christening I went to last weekend, but I haven't had time to photoshop the photos, clean them up, organize them, and so forth.  It's just been either work or eBay the last couple weeks.  Here's one picture I took (at the office, of couse).  Somebody in my building went out and got one of those "Dream Cruisers".  Isn't it gorgeous?  Wow.

On the eBay front I continue to sell and sell.  Part of my unlimited set is gone now, and I've gotten the right price for it, and I continue to identify rarities or obscurities in my collection and put them up for sale.  I am happy to report that I am at 90% of stage one and expect to make it there by the end of the week.  Cool beans!

But I have learned one important lesson about selling.  Selling in bulk may allow you to ship more product, but it definitely costs you money.  I sold a set of 8 revised dual lands back on the tenth of September for $122.50.  In preparation for the end of that auction I got another 7 dual lands together and was prepared to sell them as a batch.  But I decided not to.  As an experiment I decided to sell the 7 duals as individual auctions (here they are 1, 2, 3, 4, 5, 6, 7).  They went for a total of $139.60.

Sold as a batch, they went for $15.31 apiece.  Sold individually? $19.94.  That's 30% difference!!!  Viewed from that perspective it makes sense to sell individually unless you have a very small market (say you are selling on a table on your front lawn... maybe a couple dozen people swing by and look at the items, chances of selling them all individually are remote.)  But on eBay the market is huge, millions of potential buyers, so go for it right?

Well there *are* the eBay fees to consider.  eBay will charge you for absolutely anything you can think of and plenty of things that wouldn't even occur to you.  Back in the early days of eBay there was a listing fee and a sale fee.  The listing fee used to be 30 cents.  That was it.  Then if the item sold there was a sale fee which was this sliding scale thing that was very complicated but generally worked out to about 4% to 5% of your sale.  Nowadays everything costs money.  No longer is the listing fee flat, but it is based on the minimum bid of your auction.  If your minimum bid is a dollar, the listing fee is 20 cents.  If your minimum bid is $49.99 your listing fee is $1.20, and if your minimum bid is $59.99 the fee is $2.40.  As you can see the fee is determined inside some sort of bracketed structure.  Want to add a reserve price?  You'll be charged 1% of your reserve price.  Want to add a buy-it-now price?  There's a fee for that.  Want to schedule your auction to start later in the day or later in the week?  There's a fee for that.  And so on and so on and so on.  Sale fees aside, eBay takes a hefty chunk out of you up front just for listing the item.  To the point where it makes no sense at all to sell anything for under say, 5 dollars.  Ebay will simply eat so much of it that it becomes pointless to sell it.

And of course once your item sells, eBay charges you for a percentage of the sale price, this you would pay no matter how many items you are selling, but the listing fees are paid for each item.  My listing fees for the first dual land auction was 30 cents (I had no minimum bid, no buy it now).  The 7 individual auctions cost 40 cents apiece to list (probably because I set a minimum bid of $9.99, otherwise it would have been 30 cents).  But that is $2.80 to list 7 individual items instead of 30 cents to list one.

Then come the PayPal fees, which generally work out to about 3.5% of the sale price but which have overhead costs causing them to be at least 30 cents as a minimum.  I read an amusing story about a guy who overcharged his customer about 90 cents for shipping, and offered the customer a refund via PayPal.  So the customer got sixty cents.  Ouch.  For items worth about $20 each, the overhead is not really all that relevant, but it would be for inexpensive items.  The PayPal fees would be easier to stomach if PayPal wasn't owned by eBay.  Yes, that's right, you pay eBay to list your item, you pay eBay to sell your item, and you pay eBay to collect the payment for the sale of your item.  Yeesh.  And, to make it even more annoying, PayPal fees are instantaneously applied to any cash transferred, but eBay fees are simply billed to you at the end of the month.  That way, PayPal can charge you a percentage off of the $100 you were paid, instead of the $100 you were paid minus the eBay listing and sale fees.  I hasten to point out again that PayPal *is* eBay.  Double dipping anyone?

After correcting for eBay listing fees, eBay sale fees, and eBay PayPal fees, the group lot went for $14.20 per card, and the individual lot for $17.80.  That's still a 25% improvement in price, even after being robbed charged for eBay's fine services.  Altogether the 8 auctions sold for $262.10, which isn't bad, even if eBay took $23.92 in total fees.  My state government charges 5% in sales tax.  eBay?  9.13%.  Yikes!!!

So from now on I'm going to sell individually and try not to think about the fact that I might as well just set every tenth item on fire since I'm not going to end up with any money for it.

Looking down the notes/status on my selling/sold auctions, they read like something out of "How to be Obnoxious -- A Practitioner's Guide".  Let's run through them, shall we?

Item 1 is a very rare set which did not sell because some rather clueless sellers all decided to run auctions for the same item at the same time as mine, and undercut me.  As a result they all hurt each other and nobody made what the set was worth.  One guy had an extremely low minimum bid and got no bids.  After his auction ended, there was only one auction left to bid on, it shot way up over the first guy's minimum despite having poorer quality cards.  Thinking about that makes my brain shrivel.  Now I have to wait around for all of these shmoes to finish relisting and selling their sets so I can get a fair price for mine.  And they appear to be waiting for me.  Greaaaaaaat.

Item 2 is my most valuable set, wouldn't sell for 80% of the retail price.  I received the most ridiculous offers, promises from people who then disappeared, and got strung along by people who eventually just backed out as if surprised when I repeated stuff to them which was clearly stated in the auction description.  Meanwhile the auction has had hundreds of views and dozens of people watching it.  I've now relisted (cha-ching! extra ebay fees!) lowered the price to 70% and it still isn't selling.  Jesus.  I'm going to have to split it up and sell it in pieces (cha-ching! more fees!)... and I'll end up making way more than the current asking price.  Had one buyer offer the full price if I would ship to Spain (auction says shipping to USA only.)  He assured me up and down that it was perfectly safe and he does it all the time.  Then I told him that he would have to assume the shipping risk (i.e. item goes missing, he has to wait for the UPS refund, up to 6 months).  And suddenly it's no longer something he feels comfortable doing.  (Guess he doesn't have as much faith in the Spanish courier services after all.)  Currently I have one offer from a guy who "is trying to get the money together".  We'll see... past history is not a good indicator.

Item 3, a set so rare that only a handful exist in the world.  I set a reserve and have people who expect me to sell it to them for one tenth of the reserve.  One bidder offered to buy it for the reserve price, but insists I cancel the auction and relist it with a buy it now option first!!  (For those wondering, it cost about $9 to list it the first time, and would cost another $9 to list it again.)  Guy refuses to just bid the goddamned reserve amount.  I refused to end the auction for him.

Item 4, fixed price/best offer auction.  I accepted an offer of $150, only to discover that the buyer was from Japan (you don't get buyer's location until you accept their offer.)  Auction description says shipping to USA only.  Genius.  I recalculating the shipping charges, and they tripled.  Sent buyer an invoice and pointed out that he should have contacted me first before he bid because that's exactly what the auction description says.  I hope he doesn't back out, but I suspect he will (and I lose $5 in listing fees... cha-ching!)

Item 5, sold Aug-31.  No contact from buyer.  Invoice sent 9/1.  Reminder sent 9/4.  Still no contact.  Standard eBay grace period is 3 days from end of auction.  Going to have to send buyer a warning tomorrow and if he doesn't pay within a day after that it will be negative feedback and relist the item (cha-ching! more listing fees...)

Item 6, shipping to USA only.  Bidder from Germany pleaded with me to ship to him, and eventually I agreed and let him bid.  Then after winning the item he argued with me about the shipping costs and insisted I ship USPS instead of UPS.

Item 7, buyer asked repeatedly about card condition despite pictures of the cards being in auction description, and then took his sweet time getting his payment together after he won (took 5 days to make an instant electronic payment.)  At least he apologized for taking so long.  He'd be the first.

What next???

As you know I am selling some of my MTG collection (the oldest and the bestest stuff) to finance photographic gear.  I spent awhile researching prices so my stuff could be competitively priced--which was a little challenging because for some of the hotter stuff (the Unlimited and Arabian Nights sets) there hasn't been an auction on ebay for months on these items.  I finally figured out why.  Potential sellers are waiting for a price to beat.  After months with very little in the way of complete sets of Unlimited/Arabians, within a day of me listing my sets for auction, a bunch of other sellers crawled out of the woodwork and listed their sets for auction, all undercutting me by small amounts (typically about $50), or by setting high reserves and no minimum bid.  Since all these auctions are now running simultaneously, there is suddenly a lot of options for the prospective buyer, and since mine is ending before the others, the lower-priced auctions are going to see all the action.

So at this point I don't expect the Arabians or Unlimited sets to sell, and now I'm going to have to sit around like the other campers and wait for some other seller to come along so I can undercut him or her.

Pain in the ass.  The amusing thing is, sets this rare sell so infrequently that if a seller waited until my auction was over, he'd end up doing better than he would if they all try to sell at the same time and undercut each other.  Oh well... I guess the buyers win!