At the long term care facility at which I used to volunteer, bingo is played every other week or so. In the room in which it is held, there are six tables (numbered 1-6 in the sketch below), each holding about four players. There are also some individual seats for extra players. Each player receives two bingo cards, and bingo is usually played according to normal rules, with the exception that getting four corners on one of the bingo cards counts as a bingo as well.


Among other things, I was often responsible for distributing bingo cards to the players. I noticed that the two bingo cards I gave to a player typically carried a very similar series of numbers. I began to wonder if I could increase a player’s chances of winning if I gave them two cards with as minimal overlap as possible (and perhaps by doing so, decrease the number of balls the caller has to draw in order to reach bingo). I figured wins would occur more quickly with a diverse pair of cards, for the same reason that you gain no advantage in the lotto when having two identical sets of numbers.

Running the numbers
I made a simulation using Java, which I checked by comparing the numbers it generated with the probabilities calculated on this site. After verifying, I set up a game with thirty players, each having two randomly generated cards. I then ran 10,000 games to determine the most likely ways to reach bingo. The 10,000 games resulted in 12,051 bingos (more than one bingo can occur per game if two or more players reach bingo at the same time). The data is as follows:
running the numbers

Note the following:

  • 59% of bingos used the free space. This is especially notable since there are more ways to get a bingo without a free space (9) than there are with a free space (4). If we exclude from the analysis the numbers from four corners bingo, the percentage of bingos that did not use the free space falls from 41% to 32.5%.
  • Approximately 25% of games resulted in multiple bingos.

Is a lack of card overlap advantageous?
Next I made sure one of the thirty players had no overlap whatsoever between his cards (call him player X). All the other members had randomly generated cards, which almost always contained some overlap, usually 4 or 5 numbers overlapping between cards on average. I ran one million games. The percentage of winning bingos belonging to player X was, on average, 3.37%. To be more specific about the method, I ran 100,000 games at a time, with player X having the same pair of cards for all 100,000 games, and all other players getting randomly generated cards each match. I ran this ten times for a total of 1,000,000 games. The upshot of all this is that there is no practical benefit for a player to have no overlap between his cards, as any given player has a 1/30 = 3.33% chance of winning a game of chance involving 30 players.

Does card overlap increase time needed to reach bingo?
The above results suggest that card overlap has little effect on time to reach bingo. To confirm, I ran 100,000 games where all players had a purely randomly generated set of cards and another 100,000 where each player had zero overlap between his cards (though this doesn’t rule out the inevitable overlap among different players’ cards). The results showed no difference in number of balls drawn to reach bingo (approximately 18.0 turns in either case).

The answer to all my initial questions was “no.” Practically speaking, overlap on the cards has no effect on the outcome of the game. One reason for this is that, unlike the example of the lotto numbers, the location of the number matters. Having the same number on both cards is not always a mere repetition because the ‘value’ of a bingo number depends on the numbers around it and, also, an overlapped number may fall on a different spot in the two cards.

I think the simulation is solid, given that I verified it independently. The only possible limitation I can think of is that, in the real life situation, there are only a limited set of cards, whereas the simulation randomly generates each pair – but I do not see how that would affect my results. Another possible issue is that player X’s cards were not randomized every game, but, again, I do not see how this would be a big issue, since the balls were randomly drawn each game and all the other players had cards randomly generated each game. It does raise an interesting question though: when we want a random simulation, should we make all parts random, or is it enough to have just one part random? I can share the source code for the simulation with anyone who wants it.



  • Clare Coffey writes on the opioid epidemic, rightly pointing out the tendency to save people drowning in the river without wondering why they’re falling in, in the first place.
  • A recording of David Bentley Hart’s lecture at Fordham on American Orthodoxy has been released online.
  • Tali Sharot has an article at the Harvard Business Review about how to motivate people. The most interesting part is the studies of behavior. Apparently, they found that rewards are better at encouraging action while threats are better at discouraging bad behavior. A hospital, struggling to encourage staff to sanitize their hands before entering a patient’s room, decided to try a strategy other than warnings: “An electronic board was placed in the hallway of the unit that gave employees instant feedback. Every time they washed their hands the board displayed a positive message (such as “Good job!”) and the current shift’s hand-hygiene score would go up. Compliance rates rose sharply and reached almost 90% within four weeks…” But although Sharot is a neuroscientist, the actually biological explanations given are disappointing. It does not get much better than references to a vague ‘signal’ being triggered in some part of the brain, which does nothing to confirm the phenomena itself, since everybody already knows that the brain plays a role in behavior.
  • Adam Rogers summarizes new research by Ted Gibson and Bevil Conway on names for colors across cultures. If I understood it right, they have built on Berlin and Kay’s earlier finding that each culture divides up and names the color spectrum in roughly the same way. They show there is a possible link with how foreground objects tend to be warmer colors (which are more readily communicable) versus background which tend to be “cool”. We have less need to label background objects, like the forest or the sky, than to label foreground objects, which tend to be warm. Our names for colors, they suggest, developed this way because we develop color words inasmuch as they are useful: “as cultures get more industrialized, they get more color words. First comes black and white (or light and dark), then red. Then a bunch of others.” There is an interesting analogy here with the development of language for numbers.

St. Photius and the Nativity of the Theotokos

Thus, while each holy festival both affords the enjoyment of common gifts and lights up its peculiar glow of grace, the present feast honouring the birth of the Virgin Mother of God easily carries off the glittering prize of seniority against every competitor. For, just as we know the root to be the cause of the branches, the stem, the fruit and the flower, though it is for the sake of the fruit that the care and labour are expanded on the others, and without the root none of the rest grows up, so without the Virgin’s feast none of the those that sprang out of it would appear. For the resurrection was because of the death; and the death because of the crucifixion; and the crucifixion because Lazarus came up from the gates of Hell on the fourth day, because the blind saw, and the paralytic ran carrying the bed on which he had lain, and because of the rest of those wondrous deeds (this is not the time to enumerate them all) for which the Jewish people [τὸ Ιουδαίων ἔθνος] ought to have sent up glory and chanted praise, but were instead inflamed to envy, on account of which they perpetrated the Saviour’s murder to their own destruction. And this because Christ, having submitted to baptism, and having released men from their error, taught the knowledge of God in deed and word. The baptism was because of the nativity; and Christ’s nativity, to put it briefly and aptly, was because of the Virgin’s nativity, by which were are being renovated, and which we have been deemed worthy to celebrate. Thus the Virgin’s feast, in fulfilling the function of the root, the source, the foundation (I know not how to put it in a more appropriate way), takes on with good reason the ornament of all those other feasts, and it is conspicuous with many great boons, and recognized as the day of universal salvation [παγκοσμίου σωτηρίας].

Source: The Homilies of Photius, Patriarch of Constantinople, ed. and trans. Cyril Mango, (Cambridge: Harvard, 1958), p. 165 (Homily IX.2). It can be found in PG 102:547, where it is listed as “Homily I” instead.

St. Photius’s homily on the Nativity of the Theotokos seems to have been his most popular. It is found in more manuscripts than any of his others. This is not without reason. The homily is excellent, especially the passage I’ve quoted, with the exception where he ascribes Christ’s murder to “the Jewish people,” which led them “to their own destruction.” I think St. Photius speaks this way because he has in mind the destruction of Jerusalem. Just as the destruction fell on the Jews as a whole, so its cause is attributed to them as a whole, even though this is obviously false. The condemnation of the Jews here only extends to that generation, and not to all time. This, anyway, is true of Eusebius in his Church History, 3.5.3, and I see no reason not to extend it to Photius. All this is, admittedly, speculation, and I think it’s necessary to read all of Photius’s work before I can make a confident judgment about his understanding of the Jewish people. In any case, it remains a tragic expression due to the later evils that such ways of speaking would give rise to.

Authenticity of ‘De anathemate’

Upon further investigation into the beautiful homily attributed to St. John Chrysostom that I translated a while back (the one against anathematizing others), I have discovered that its authorship has been more contested by current scholars than I had thought. I was content with the fact that a Chrysostom translator for the CUA Fathers of the Church series had accepted it as genuine. But, I’ve just discovered, other authorities hesitate. According to Wendy Mayer in her book The Homilies of St John Chrysostom (at least what I could glean from Google Books’ snippet view), the homily is “now no longer assigned to Chrysostom” (p. 75). It is listed in the Repertorium pseudochrysostomicum (no. 448), and Lampe attributes it to St. Flavian I of Antioch. Perhaps Lampe is following the argument of Cavallera, who attributes it to Flavian based on two passages that imply it was preached by a bishop of Antioch. Firstly, in the second section of the homily: “Do you know what a holy man once said, who, before us received the διαδοχῆς of the apostles and was judged worthy of martyrdom?” The man referred to is St. Ignatius of Antioch, who was a bishop of that city, with διαδοχῆς meaning something like ‘heritage’ or ‘inheritance’ or perhaps ‘succession.’ Secondly, in the fourth section:

Do we not make public supplications for the ignorance of the people? Are we not obliged to pray for our enemies, for those who hurt us and persecute us? Right now I am fulfilling a duty of my ministry in exhorting you; χειροτονία is not a source of pride, it gives no right to despotism: we have all received the same Spirit, we who are called to the title of adopted sons: those to whom the Father has given power, have it only to serve their brothers according to their power.

On this Cavallera comments:

This allusion to the liturgy that he celebrates and to the χειροτονία which he received is clear after the preceding passage. We are not dealing with the ordination of a mere priest but rather of a bishop, to which the word χειροτονία especially applies in Christian usage. Priestly ordination is more commonly designated by the term προχειρίζω (cf. S. Jo. Chrys. Sermo cum presbyter ordinatus, PG 47, 693).

After concluding his argument that a bishop of Antioch had written the homily, Cavallera adds arguments for why the date of the homily must be such that it falls under the episcopate of Flavian. Earlier authors offered stylistic arguments for why the work could not belong to John Chrysostom. All of these arguments I am unable to evaluate, due to my very poor knowledge of Greek and the works of St. John Chrysostom.

If anyone has access to the whole Wendy Mayer book, I would enjoy seeing her full evaluation of the homily’s authorship. Also I would enjoy seeing anything on it from Sever J. Voicu, who seems to be the expert on pseudo-Chrysostom. In any case, even if the sermon is not by St. John Chrysostom, this is certainly no reason to doubt the truth of its message, especially given that, granting the wrong attribution, there is good reason to suspect it is actually by another saint, namely Flavian of Antioch.

Ss. Alexander, John, and Paul, Patriarchs of Constantinople

Today is the feast-day of three patriarchs of Constantinople, Alexander, John, and Paul. St. Paul was the patriarch from 687-693 and presided over the Council in Trullo. St. John is thought to be the patriarch from 562-577. St. Alexander was patriarch during the Arian controversy, and is perhaps best known for his presence in the story of Arius’s death (source):

When Arius had deceitfully professed allegiance to the Council of Nicaea, Saint Alexander, knowing his guile, refused to receive him into communion; Arius’ powerful partisans threatened that they would use force to bring Arius into the communion of the Church the following day. Saint Alexander prayed fervently that God might spare the Church; and as Arius was in a privy place relieving nature, his bowels gushed forth with an effusion of blood, and the arch-heresiarch died the death of Judas.

However, it would be a mistake to conclude that the Church takes pleasure in the death of Arius. St. Athanasius relates essentially the same account in his letter to Serapion, but gives this note on it:

Such has been the end of Arius: and Eusebius and his fellows, overwhelmed with shame, buried their accomplice, while the blessed Alexander, amidst the rejoicings of the Church, celebrated the Communion with piety and orthodoxy, praying with all the brethren, and greatly glorifying God, not as exulting in his death (God forbid!), for ‘it is appointed unto all men once to die’ (Heb. 9:27), but because this thing had been shown forth in a manner transcending human judgments. For the Lord Himself judging between the threats of Eusebius and his fellows, and the prayer of Alexander, condemned the Arian heresy, showing it to be unworthy of communion with the Church, and making manifest to all, that although it receive the support of the Emperor and of all mankind, yet it was condemned by the Church herself.


  • Tikhon Alexander Pino has recently uploaded a critical appreciation of the legacy of St. Gregory Palamas among the 20th century ‘neo-Palamites.’
  • The opening of a hospital by the Romanian Orthodox Church.
  • The Mayweather-McGregor fight finally took place. The sensational event brought to my attention some interesting perspectives on boxing: the first from The Hedgehog Review (2015) about the personal benefits of boxing and physical competition, and the second from Sports Illustrated (1962) about moral theology and professional boxing:

    Among these characteristics there is the element of a career involving a whole series of fights with cumulative effects. There is the admitted effort of most professionals to win by a KO—or at least a TKO—rather than by decision. There is the medical report of injury, particularly to the brain. There is the synthetic notion of courage wherein confession of injury followed by retirement from a fight invites derision by a crowd that enjoys a beating, clamors for the kill and lustily boos evasive tactics. There are the undeniable benefits that boxing has brought to the lives of many individuals. There are television contracts which create severe scheduling demands; there are boxing commissions and control groups. Finally, there is a specific set of rules. Professional boxing involves more and longer rounds, lighter gloves and sometimes different scoring criteria. These are the things the moralist attempts to evoke with the phrase professional boxing as it is today.

  • At Pemptousia, Abbess Thekla gives an interview on the history of Orthodoxy in the Democratic Republic of Congo.


  • The Russian Orthodox Church is finally nearing the completion of its catechism. You can read a draft (but only in Russian) here. The first three parts of the draft are open to comment by theologians, but the last three parts are not. The final part of the draft part (part VI) defends Orthodox ecumenism. An article from an anti-ecumenist website is also available.
  • Elizabeth Bruenig has a post on St. Augustine.
  • Fr. Christiaan Kappes has a new paper expanding on St. Mark of Ephesus’s understanding of the papacy:
  • Eugenikos’ vision can be gleaned from his opusculum: Latins’ Dissension with Roman Solutions [on the Pope] (c. 1437), as from works and speeches: (i.) Despite high honors and canonically recognized apellate power, from Chalcedon to Trullo, popes (e.g., Honorius) have personally erred in the past, (ii.) It is true that pope Agatho professed inerrancy of the Petrine see at the Sixth Synod, but this must be relegated to the faith of Peter and the Church (not a personal charism), (iii). prohibitive canons at Chalcedon on additions to the Creed are correctly interpreted by the Greeks as shown by Pope John VIII’s Photian synod, (iv.) A pope that reverses papally and dogmatically ratified canons goes against the patristic consensus (πατερικὸν φρόνημα), (v.) Such an act introduces ecclesiastical dissension and innovation (καινοτομία), (vi.) Papal decrees (jurisdiction) for the universal church may never violate subsidiarity, (vii.) Orthodox must dutifully resist and oppose a pope (as Honorius).

  • It appears that in addition to the New Jerome Biblical Commentary, the New Jerusalem Bible is also receiving a significant update.
  • ‘To reduce gender biases, acknowledge them,’ writes Debbie Chachra on the James Damore Google fiasco: “I have lost patience with arguments from people who think they are saying ‘what everyone is too afraid to say’ without recognizing that they are simply repeating what women like me have heard throughout our lives.” Outside of these sharp words, a very gracious article.