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Below are 20 journal entries, after skipping by the 20 most recent ones recorded in Mark Gritter's LiveJournal:

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Monday, January 26th, 2015
7:29 pm
Tintri wins Infoworld "Technology of the Year" 2015
The Tintri VMstore was one of 33 winners selected by Infoworld for their 2015 Technology of the Year. It's nice to get some mainstream(-ish) recognition and be put in the same category as cool consumer tech like the iPhone 6 and hot software technologies like Neo4j or Apache Storm. There were just four pieces of enterprise hardware on the list: the Tintri VMstore, HP Moonshot (that thing is a beast), Dell PowerEdge R730xd, and ioSafe 214 (which I think is more of an SMB product.)
Sunday, January 25th, 2015
1:06 am
Divisor Sums
This post may be difficult to read because LiveJournal doesn't support any of the nice Javascript math rendering libraries, and I'm too lazy to copy and paste images.

It turns out there is a nice way to calculate sum_{k=1}^{n} d(k^2), where d(x) is the "number of divisors" function (also called sigma_0 or tau or d_2) The key is that we can use Dirichlet convolution (denoted by *) to write d(n^2) = d * μ^2 (where μ is the Mobius function). Then some algebraic magic turns the sum into a sum of sqrt(n) terms, namely sum_{a <= sqrt(n)} μ(a) D_3(x/a^2). And D_3(x) is the sum of the number of ways to divide numbers up to x into three factors, which can be calculated efficiently by counting lattice points under a hyperbola. See http://math.stackexchange.com/questions/133630/divisor-summatory-function-for-squares

I'd like to do this for sum_{k=1}^{n} d(k^3) but the same logic doesn't work. I've found three convolutions so far:

d(n^3) = d * μ^2 2^{ω(n)} where ω is the number of distinct primes dividing n. Can't seem to get rid of that term, it mucks everything up.

d(n^3) = 1 * 3^{ω(n)} is even worse, it would require some entirely new trick.

d(n^3) = F(n) * μ^2 where F turns out to be a multiplicative function defined by

F(p^{2k+1}) = 3k + 3
F(p^{2k}) = 3k + 1

for example F(2) = 3, F(4) = 4, F(8) = 6, F(16) = 7. And of course its convolution with 1 is also multiplicative, but instead of ending up with something nice like a sum of divisor functions it's a sum of this wierdo thing. Neither F nor F*1 nor their partial sums appear in OEIS.

So I could write this as an algorithm but it would require factoring quite a few terms, and if that was feasible we could just calculate the sum directly without any rewriting. I'm wondering if there is some general technique I just don't know about, or whether I need one more insight. Maybe F can be expressed in terms of divisor functions in some useful way.

(The reason I believe it's feasible at all is because the calculation appears on a programming problem site, though not yet solved by anybody... Several people have solved the d(n^2) version.)
Tuesday, January 20th, 2015
10:04 am
I've read some of the literature on programming errors, about what sorts of mistakes programmers are likely to make and some analysis of why. But so far I haven't really seen analysis of more basic problems like that illustrated in today's WTF, where the programmer wrote an array to a file in order to determine the length of the array:
private static int GetPhysicalFileSize(byte[] Imgbyte, string InputObject, string OutFolder) {
  int PhysicalFileSize = 0;
  try {
      ExtractImage(Imgbyte, "Photo_" + InputObject + ".jpg", OutFolder);
      var Outfile = OutFolder + "\\Photo_" + InputObject + ".jpg";
      if (File.Exists(Outfile)) {
         FileInfo fi2 = new FileInfo(Outfile);
         PhysicalFileSize = (int)fi2.Length;
      if (File.Exists(Outfile)) {
  } catch (Exception ex) {
  return PhysicalFileSize;
There are several things that could be going on here:

* The author might not be making the connection that "size in memory" is equal to "size in file". This is actually a good thing to wonder about, given that (say) a UTF-8 encoded text file does not have the same size in bytes as its memory representation with wide characters. Better safe than sorry?

* The author might not know how to take the length of the array. Unlikely, but possible.

* The author is exhibiting poor planning skills. Rather than taking the shortest possible route to the answer, the programmer instead engages in some sort of depth-first search across operations he or she knows how to implement. (Given the second example in the article, this seems the most likely explanation.)

It would be interesting to read analysis of these more "basic" errors--- maybe in the educational literature rather than systems literature? Any ideas?
Monday, January 19th, 2015
4:00 pm
Books of varying degrees of success
Donald Knuth, "Selected Papers on Fun and Games": This is the last (and definitely my favorite) of his selected papers series published by CSLI. Not everything here is a winner, but there is a lot to love on art projects, word games, vanity license plates, mathematical Christmas cards, and knight's tours. Donald Knuth has been doing "I bet I could write a computer program to figure that out" for a very long time. Also includes a reimplementation of the classic "Adventure" in Literate Programming (C).

Tony S. Daniel, "Batman: Detective Comics" vol 1 and 2.
John Layman, "Batman: Detective Comics" vol 3 and 4.

I liked volume 1 and 3 which I picked up secondhand, so I asked for more. Volume 2 was an absolute stinker and made no sense. (It featured multiple Large Hadron Coliders? And time travel something something?) Layman engages in much more comprehensible and enjoyable storytelling. I think, though, that I am just as glad not to be pulling this series weekly. I really enjoy self-contained graphic novels more.

Ed Brubaker, "Gotham Central" book 1: Much better.

Marko Kloos, "Terms of Enlistment". I've been wanting some military SF other than Bujold, Heinlein, and Scalzi. This is not it. It's not about anything. It has three disconnected set pieces, none of which are really followed up in any way. The character is not particularly sympathetic, nor does he grow in any way. I am supposed to treat the appearance of aliens as a reason to get the next book, I guess? Not recommended. (Previous failures in this department :"Fortune's Pawn" by Rachel Bach and "Star Carrier" by Ian Douglas.)

Ann Leckie, "Ancillary Sword". We need more science fiction about tea. (And, look, it's military SF that's actually about something!) A very good follow-up, recommended.

Richard von Glahn, "Fountain of Fortune: Money and Monetary Policy in China 1000-1700". This book was footnoted in "Debt: the first 5000 years" and it is easy to see why. The author talks about the role of money in Imperial China, and how the purposes economists state for money (store of value, measure of value, means of exchange) were frequently separated into different media. Mainly an argument about the import of silver specie into China; he presents evidence that not only was American silver a sideshow (most came form Japan) but it shouldn't be interpret as a balance of trade issue so much as a commodity. There were a lot of threads I wish he'd followed up on--- like, the Chinese had their own bullionists who insisted that money was wealth even if it was locked up. So the empire would mint a lot of bronze coins and then lock them up in the treasury, while the provinces would complain about their not being enough coins to go around.
3:39 pm
VAT -- Comments too long for Twitter
So the EU switched where VAT is collected for digital service from the seller to the buyer. This is not really shocking, and removes a big incentive to play locale games.

I understand the problem that small businesses may have some difficulty adjusting. But I question how big a hurdle it really is, and unfortunately I think buyer-side taxation ultimately makes more sense. (We lost the fight to have the laws applying to an internet transaction be the server side a long time ago, see: porn, online gambling, banking, etc.)

* "78 tax rates across 28 markets"? Cry me a river, the US has what, thousands of different sales tax regions?

* If somebody is selling on the internet, they really really should not be running their own shopping cart and billing software. Once you cross this hurdle, the software providers manage the extra complexity, not you. (Yes, it sucks paying an intermediary.)

* If you're calculating it by hand, an extra lookup in your cumbersome manual process doesn't kill you. Making quarterly payments to a lot of different tax authorities may hurt. I think we'd be better focused on solving that problem than fighting against the tide here.

I really don't get what "simply cannot comply with this law" means. Sole traders can either do the extra work or not sell outside their borders, but what exactly is the high cost of compliance here? Is that too US-centric an attitude?
Friday, January 2nd, 2015
11:39 am
"Fountain of Fortune" and arbitrage
The conventional story of East-West trade is that China accepted only bullion from the Atlantic economies because the Europeans had nothing else to offer that was of interest. But that's not all that's going on.

China had an immense hunger for silver, from all of its trading partners. There were several reasons for this. One was that silver was used as a store of value. After numerous depreciations in coin and paper, nobody trusted the government's currency, and there was never enough silver. Another was that silver wasn't subject to import duties! (The various import duties by themselves discount the notion that there was nothing the Chinese wanted to import...)

But the biggest reason to import silver to China is that China had a vastly different gold-to-silver exchange ratio than the rest of the world:
Chinese merchants did not simply pursue bullion; they sought one particular metal--- silver. The source of profit from silver was, of course, the high price silver commanded in China. The gold/silver ratio in China had drifted slightly downward from its historical peak of 1:4-5, achieved in the late fourteenth century, to 1:6 by the early sixteenth century. In contrast, the gold/silver ratio hovered around 1:2 in Europe, 1:10 in Persia, and 1:8 in India... --- Richard Von Glahn, "Fountain of Fortune: Money and Monetary Policy in China, 1000--1700", p. 127

Von Glahn argues that most of the silver imported into China actually came from Japan (once Japan opened significant silver mines), not the Americas. Some of this trade was carried out by Europeans, but by no means all.

I received "Fountain of Fortune" for Christmas and it's an interesting read. Much of it is about how the three conventional functions of money (store of value, measure of value, and unit of exchange) were served by different media for centuries. There are a variety of reasons for that--- one is that China simply couldn't mint enough bronze coin! A lot of Chinese bronze coin (like, entire shiploads) was imported into Japan for use there. But the perspective of the American bullion trade as merely a sideshow in a giant gold-silver arbitrage scheme is also an interesting bit of history to read too.
11:07 am
Exploring in Elite: Dangerous
Elite: Dangerous doesn't get enough mileage out of its gorgeous planets. There's little reason to go visit them. Trade is done through stations, mining is done in asteroid belts (and ring systems, admittedly), and exploration can be done from 5+ light-seconds away from your target.

Anyway, here's some eye-candy of the first earthlike world I discovered, officially designated "Col 285 Sector AF-P c6-1 4". I love the large inland seas:

Screenshot (7)

"Four" is a cold world (-11 degrees Farenheit) and doesn't yet have oxygen in its atmosphere--- no cyanobacteria?

Screenshot (8)

Planets four and five orbit each other. Five is an ice world with giant fissures and some volcanic activity discoloring the surface:

Screenshot (9)

Unfortunately, Elite doesn't allow you to name (in-game) the planets you discover. In an online world there is, of course, potential problems with vetting large numbers of names for indecency.

I really hope that Frontier Developments finds ways to make better use of the hard work they put into solar system models. It'd be great to name the planet, get a detailed surface map, perform additional surveys, and launch a colonization expedition. Realistically, of course, all these things would take years if not decades. (And the game doesn't even have stations that are under construction within settled space!)
Friday, December 12th, 2014
11:31 pm
More "Elite" ramblings
I am pretty happy with "Elite: Dangerous" so far, in its prerelease state.

The economic system is fairly well thought-out, and I am surprised at the number of people who complain about problems making money. It may be that players are relying on the in-game import/export tools too much. These can be misleading because they operate on a system, but the economy actually varies by planet/station. (So in-system trade can be profitable, too.):
* Everybody needs food, clothing, and medicine. Rich economies want luxury items.
* Agricultural planets produce food items and need agricultural machinery. They also have at least some luxuries, usually beer.
* Industrial planets produce clothes, medicine, and machinery of various sorts, and need input materials such as metals, cloth, and semiconductors.
* Refining stations turn minerals into metals and other inputs.
* Extraction stations industries produce minerals.
* High-tech planets produce luxuries, automation items for all the above, and need computers and input materials. They also produce weapons.
* There are some niche/specialized economies I haven't yet fully explored: "Imperial", terraforming, service, tourism. Also some planets have a blended economy.

It's not hard to set up loops of various sizes:

Agricultural <-> Industrial works well
Extraction -> Refining -> Industrial -> Extraction is a little trickier because not all ores are refined at all stations, but you can easily learn what works (and their trade diagram has gotten better at indicating what minerals exist where.)
Agricultural -> Industrial -> High Tech -> Agriculture works well with a rich agricultural world. Can substitute in Refining instead of Industrial.

Some of the gamma players seem to really not know what to do with themselves in a truly open world. There's not a plot to hop back onto when you're tired of the side quests. However, the political simulation (factions get influence based on user input) provides the seeds of good endgame play, and can provide near-term goals for starting players too.

The final complaint I don't really get is that the game is "incomplete" without the ability to land on planets (which was part of "Frontier"). I can totally understand why that got cut as a release feature, and I don't understand why players are so attached to it--- it provides some more landscape but there is plenty to go explore without actually visiting the surface. If this feature were present, it wouldn't add to the gameplay in any way.

The thing I think is most likely to cause problems is that any of the "evil" options are so heavily punished that players can't even try them out. Piracy or assassination or just being caught with salvaged ("stolen") or smuggled goods puts you on a downward slope with the local government, and it seems very hard to get back into their good graces. (ED doesn't have alternate characters either to let users explore different ways to play.) This is fine from a simulationist perspective, but from a player perspective it can feel like those options are locked off. On the other hand, I think this is something where paying attention to the faction system and local politics can really pay off. If you go pirate in an independent system, the Federation doesn't care.

Compared to "Frontier", combat is a lot more fun and varied. Frontier sacrificed playability for adherence to Newtonian physics. Elite: Dangerous gives you enough control to do some fancy maneuvering, but puts a speed limit in place so you don't just whip by your opponent at high speed. There are also a bunch of customization options and different sorts of weapon systems to try out.

Overall, the game looks good, plays pretty well, and is a quite fun update for anybody who liked the original Elite (or even Frontier.)
Wednesday, November 26th, 2014
11:34 pm
Future Calendars
I have started playing the "Elite: Dangerous" prerelease version (the gamma release). One choice the developers made is to continue to use real-time calendar dates, just advanced to the year 2300. That is, if you're playing on November 25th then the game reports the 25th of November 2300.

This works fine, except for leap years (the day of the week isn't shown.) 2016 is a leap year, but 2302 is not. Presumably David Braben is planning to continue to support the game for at least two years, so why not start with 2314? Then the problem doesn't occur until 2100/2400.

What about leap seconds? Well, the median solar day is currently increasing by about 1.7ms per century. The net error in 300 years should be around 280 seconds, big enough to notice! But you can just pretend that the ITU (or successor organization) made up its mind to abolish leap seconds, while strangely keeping the Earth calendar.

I suppose future-ITU could also be deemed to have change the leap years for the sake of game consistency as well. :)
Monday, November 24th, 2014
12:22 am
Tintri included in Gartner's "Magic Quadrant" for General-Purpose Disk Arrays.
Gartner is a well-known research and advisory firm in information technology. Their most popular reports use Gartner's "magic quadrant" methodology to compare firms based on their ability to execute and vision for the technology.

Garter's 2014 report on general-purpose disk arrays includes Tintri for the first time! We are identified as a leader in the "visionary" quadrant based on our development of smart storage arrays that see, learn, and adapt. Customers see Tintri's benefit in lower operational costs and consistent performance. We placed higher on the vision axis than Nimble, Tegile, and even long-established players like IBM.

Although Tintri only works in virtualized environments, Gartner correctly notes that more than half of server deployments are virtualized. Storage for virtualization is general-purpose storage.

Items that hold us back from a higher "execution" score include our status as a non-public company, better scale-out stories through automatic load-balancing, and the overall scale of our business--- we're still small, but growing.

More information in Tintri's announcement page here: http://info.tintri.com/magic-quadrant.html

Tintri also launched a complete new platform line this month. The Tintri VMstore T820, T850, and T880 come with on-disk compression, a host of new software features, and a shiny Portal-esque bezel: http://www.tintri.com/products/tintri-vmstore
Wednesday, October 15th, 2014
11:48 pm
Druid Deck
I have been trying to build a Hearthstone Druid deck but the class hasn't been making a lot of sense to me. I'm finally having some success in non-ranked games with the following:

2x Innervate [0]
2x Naturalize [1]
2x Mark of the Wild [2]
2x Power of the Wild [2]
2x Wild Growth [2]
2x Echoing Ooze [2]
2x Faerie Dragon [2]
2x Healing Touch [3]
2x Emperor Cobra [3]
Ironfur Grizzly [3]
Swipe [4]
Keeper of the Grove [4]
2x Druid of the Claw [5]
Starfire [6]
2x Ironbark Protector [8]
Kel'Thuzad [8]
Onyxia [9]

There are a lot of nice combos at work here. But when it doesn't work out I find myself card-short and unable to respond to a deck with more depth. Thinking of ditching some of the mana cards in favor of some higher-value cards, or more drawing cards.

A Druid deck I saw that was sort of surprising was all +spell damage minions, which powered Moonfire/Swipe/Starfire/Wrath.
Saturday, October 11th, 2014
1:33 pm
I need a better Go app [crossposted on G+]
"Go Free" successfully beats me 2 out of 3 times at half-difficulty and a two-stone handicap. (What can I say, I suck.) But it's still occasionally capable of gross stupidity like this position, where it marks the upper-left corner as dead (?!?) and doesn't recognize seki in the lower-left.

14 - 1(1)
Sunday, October 5th, 2014
12:49 pm
On reading Joe Abercrombie
Thirr the Golden stalked across his tower's roof.

"Twenty-five years! And not one of these stories have taken root!"

His secretary kept silent.

"Princes swapped at birth become cowardly shepherds. Betrayed mercenary captains simply vanish. Heroes condemned to the galleys die of dysentery."

"I thought the child raised by wolves was off to a good start, master?"

"I just checked her worldline. She broke her ribs tackling an elk, and caught a fatal case of pneumonia shortly thereafter. My star-crossed lovers failed to show up for the barricades. Thieves and gladiators appear singularly uninterested in overthrowing tyrants. And don't get me started on talking cats, mirrors, and swords! The last ended up in the possession of an insane brute who already heard voices in his head. If I could get the epic geas to take hold but once..."

The secretary, suddenly close, murmured, "perhaps a trusted aide turned to treachery?"

Thirr vainly attempted to summon flight as he fell.
Monday, September 22nd, 2014
9:52 am
Compiler writers aren't actually idiots
Occasionally I see 0-based indexing in programming languages justified in terms of efficiency of not having to subtract one when performing array lookups. For example, the C code
x = my_array[i]
translates into something like
movl (%ecx, %esi, 1), %eax
directly (where ecx holds my_array and esi holds 'i'), while presumably the Pascal equivalent
x := MyArray[i]
has to adjust 'i' by whatever the start index is (usually but not necessarily 1.) But there's no ironclad rule that says the implementation of 'MyArray' has to point at the actual address of the start of the array. The compiler writer can arrange things so that the exact same assembly is generated, can't she? (Except that Pascal performs bounds checking...)
Sunday, August 31st, 2014
7:15 pm
VM granularity and data reduction
At VMworld I heard this idea a couple times, once from an analyst and once from somebody with interest in vVols:

With per-VM controls, an administrator could turn off compression and deduplication for VMs that don't benefit from it, and thus improve overall system performance.

In my mind, this is something that "sounds good" but falls apart based on closer examination.

1. Users generally don't do a good job telling whether compression and dedupe are effective on a VM.

One example provided in support of the idea is applications which consist of already-compressed data like video. Or, perhaps the VM is running with guest-level full disk encryption turned on. The OS could also be doing compression and dedupe within its own file system.

But, at least two of these examples are flawed. While compressed video files are unlikely to be compressed further, the VM contains OS files, file system metadata, and even video metadata, all of which compress well. Even the video portion may be duplicated if the guest OS performs any swapping, runs a log-structured file system, or contains redundant copies of the same video. A whole-VM or even whole-disk decision could abandon significant data reduction.

I would also argue that the guest OS generally should not be configured to do its own compression and deduplication. You will get worse behavior from turning on compression on all the individual VMs rather than letting the storage compress and dedupe globally, unless the storage controller is so short on CPU power that it can't do a good job, which brings us to the next point:

2. Poor inline compression and dedupe does not significantly affect storage throughput and latency.

I can tell you from personal experience at Tintri that we had far more problems with data that has high compression and dedup than data with low ratios! Data that doesn't compress nor dedupe is easy to accommodate; it doesn't take any more CPU cycles to fail to compress than it does to compress. But data that is extensively deduped requires more metadata operations than normal writes. Data that compresses well may leave a flash segment unfilled when the non-compressed version has exhausted non-volatile storage. (We don't compress data on the way to NVRAM, in part to keep latency low. This means that dedupe and compression don't have any first-order affects on write latency anyway--- only second-order effects based on resource consumption later in the write pipeline.)

Consider what turning off inline data reduction means. It does mean we would get some CPU savings from not compressing a block--- but this cost already has to be low to afford to do it inline, tens of thousands of times per second. There is no savings in checksumming the block, writing it to flash, modifying file system metadata, or garbage collecting the block later. If the system is properly engineered to perform inline data reduction, "turning it off" affects just a small part of the cost, one that may not even be the bottleneck for a particular workload.

However, if compression and dedupe are the bottleneck, the user is unlikely to be sure this is the case, which is my final point:

3. If there is a benefit to be gained, the storage system should do it automatically.

Tintri's goal is to build storage that "sees, learns, and adapts." If a particular VM does not compress or dedupe well, and there would be an overall system benefit in turning compression off for that VM, then the storage system should self-tune to achieve the benefit. Compare:

A. Administrator migrates a problem VM onto her storage
B. The problem VM causes bad storage behavior, which the storage system displays to the administrator in some comprehensible fashion (if she's lucky)
C. The administrator notices the problem, correctly diagnoses the underlying cause, and performs the correct configuration change without error (most of the time.)
D. Repeat it all over again for the next VM. Or the next 100 VMs. Or the next 1000 VMs across multiple storage appliances. If she can dedicate the time, compared to all the other things on her priority list.


A. The storage notices that a VM has unusual behavior and tailors its treatment of the VM to match.

Which world would you rather live in?
Wednesday, August 13th, 2014
8:12 pm
Sub-quadratic by a tiny, tiny bit
I was reading a post on the 3SUM problem, and a commenter pointed to a new result: "Threesomes, Degenerates, and Love Triangles" which produces a new bound that is just a tiny bit better than existing results.

The 3SUM problem is: given a set of numbers, does there exist a triple a,b,c such that a+b+c = 0? For real numbers, the best existing algorithm is O(N^2) and a lot of conditional bounds on other problems can be derived from the assumption that that's the best possible.

The new paper introduces an algorithm that is O(N^2 / (log N / log log N)^(2/3) ).

How much better is that? Well, at N=1000000, the numerator on that fraction is just 2.773. To get to a factor of 10, you need N to be about 10^75. In other words, for all practical purposes, the algorithm is useless--- rather than trying to get it right, you're better off tuning your implementation of the simple algorithm. But it's a potentially interesting advance because they also showed a better decision-tree bound than previously known, and got "subquadratic", but unfortunately not O(N^(2-e)).
Wednesday, August 6th, 2014
12:34 pm
Tintri 2Q2015
Tintri switched to a February-based fiscal year so we just finished the second quarter of fiscal 2015. We had the last all-hands meeting in our old office, and are scheduled to move on August 15th to our new location (just two buildings away, still on Ravendale Drive in Mountain View.)

I can't talk financials (but they're in good shape) but I can highlight our growth in people! Currently we're at 283 people, of which 121 are new this year. Our goal is to get to around 340 by the end of the year. (It's been mind-boggling: at the end of 2011 we were still only at 52.)

About 35% are in engineering and 40% in sales. We're still hiring aggressively, so please let me know if you or somebody you know are interested in joining a great pre-IPO company! We're looking for people with experience in storage, virtualization, and "the cloud"--- but have also hired great people outside those areas too.

Tintri is a platinum sponsor at VMworld US this year; I'll be out working the booth at the end of the month.

ETA: Good coverage from Forbes: Tintri's 140% Growth Threatens Shrinking NetApp. Ken also mentions we are getting 70% gross margins. A few other tidbits: 65% of revenue is US, 35% rest of world; and we have 5 customers in the top 15 Fortune 500 companies.
Sunday, July 27th, 2014
7:23 pm
Oh, micro-stakes poker
I played a 0.01 BTC heads-up Razz SNG today. The opponent seemed fairly competent, but the final hand made me question my own judgement of him.

I have 3860 chips with (68)9 and he has 2140 with (xx)K, at the 30/300/600 level. I raise, naturally, and Villain re-raises. I three-bet and it takes a few more raises to get him all-in.

Is there any hand I can reasonably have here that makes him even break-even? He had (A2)K which is fine for a steal defense, but a shove? Why not wait until 4th? Perhaps he was just tired of playing.

Razz (7-card Stud A-5 Low): 500000 sampled outcomes
cards         win   %win    lose  %lose  tie  %tie     EV
Ad Kd  2d  185344  37.07  314538  62.91  118  0.02  0.371
6c 9h  8h  314538  62.91  185344  37.07  118  0.02  0.629

I would need to have something like (JT) or (Q8) down before he becomes a favorite.
Saturday, July 12th, 2014
2:08 am
"The Avengers: Earth's Mightiest Heroes" Drinking Game
One drink if:
* Somebody holds Hornet in their hand
* Hornet is knocked unconscious
* Hornet's stingers have no discernible effect
* Ant-Man mopes about not wanting to hurt anybody

Two drinks if:
* Hawkeye runs out of arrows
* Somebody else brings extra arrows for Hawkeye
* Hawkeye's special arrows are easily ignored (not countered)
* Ms. Marvel appears out of uniform

Three drinks if:
* The Avengers forget that fake-Captain America betrayed the Hulk and he's still captive somewhere
* The Hulk changes back to Bruce Banner
* The Hulk is wearing pants that aren't torn
* Thor declares that something will "end now"

Four drinks if:
* Pepper Potts appears (extra drink if Tony's not in the scene)
* Iron Man wears an alternate suit of armor
* Iron Man refers to financial or emotional problems
* T'Chala shows up without his mask
Friday, July 11th, 2014
1:17 am
Ellsberg's Paradox
I'm reading Jordan Ellenberg's "How Not to Be Wrong". He describes a paradox I don't recall seeing before. Here are four bets on drawing a ball from an urn, which has 30 red balls and 60 balls that are a combination of black and yellow (in some unknown proportion):

RED: the next ball is red
BLACK: the next ball is black
NOT-RED: the next ball is black or yellow
NOT-BLACK: the next ball is red or yellow

Survey time--- if you can bet $100 on one of the first two if you are offered a $100 prize based on one of the first two conditions occurring, which one do you prefer? How about your preference among the second pair?

If offered RED or BLACK, I prefer

I am indifferent

If offered NOT-RED or NOT-BLACK, I prefer

I am indifferent

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