This week the question is: Can You Play the Favorite?

March Madness—the NCAA basketball tournament—is here!

The single-elimination tournament consists of 64 teams spread across four regions, each with teams seeded 1 through 16. (In recent years, additional teams beyond the 64 have been added, but you needn’t worry about these teams for this week’s puzzle.)

Suppose in any matchup between teams with seeds M and N, the M-seed wins with probability N/(M+N), while the N-seed wins with probability M/(M+N). For example, if a 3-seed plays a 5-seed, then the 3-seed wins with probability 5/8, while the 5-seed wins with probability 3/8.

In one of the brackets, the top four seeds remain (i.e., the 1-seed, the 2-seed, the 3-seed, and the 4-seed). If case you’re not familiar with how such brackets work, at this point the 1-seed and 4-seed face off, as do the 2-seed and 3-seed. The winners then play each other.

What is the probability that the 1-seed will emerge victorious from this region?

And for extra credit:

As before, the probability that an M-seed defeats an N-seed is N/(M+N). But instead of 16 teams in a region, now suppose there are 2^k teams, where k is a very large whole number.

The teams are seeded 1 through 2^k, and play in a traditional seeded tournament format. That is, in the first round, the sum of opponents’ seeds is 2^k+1. If the stronger team always advances, then the sum of opponents’ seeds in the second round is 2^k−1+1, and so on. Of course, stronger teams may not always advance, but this convention tells you which seeds can play which other seeds in each round.

For any such region with 2^k teams, what is the probability that the 1-seed emerges victorious from the region?

Highlight to reveal (possibly incorrect) solution:

Program .

In order for the 1-seed to win, they have to beat the 4-seed and the winner of the 2-seed/3-seed match. The first win has probability 4/5. The winner of the other match is the 2-seed with 3/5 probability and the 3-seed with 2/5 probability. The 1-seed is victorious in this 2nd match with probability 3/5 * 2/3 + 2/5 * 3/4 = 7/10. Combined with the 1st win, the overall probability for a winner 1-seed is 4/5 * 7/10 = 14/25 = 56%.

My program confirms this.

And for extra credit:

There’s probably a smart way to combine all the above fractions into a sum, that then turns into, I don’t know, sin(2^k). 😁 I will however stick to my program, fiddled (!) a little to let k grow and see what happens. The probability quickly converges to 51.6%.

Recently I’ve been reading my way through one of those bundles. In this case so, so many books by John Scalzi. A lot of reasons for that. The bundle was cheap. And I liked Redshirts, and Fuzzy Nation, and some other stuff here and there. Oh yeah, Old Man’s War. It would be nice to read the rest of the series.

But the language is increasingly rubbing me the wrong way. So many words could be deleted. A lot of sentences could be shortened. And in some cases, a sentence is simply constructed wrong or represents a falsehood. (Disclaimer: My English isn’t perfect either. But I’m not a millionaire author.)

So, just for funzies, let’s look at a random segment of text.

The Last Emperox. Prologue. (Hopefully a link to the Kindle preview.)

ooo

… he only got as far as saying “I,” and really only the very first phoneme of that very short word…

(“I” only has 1 phoneme.)

ooo

… the surfaces of the aircar’s passenger cabin, Ghreni Nohamapetan, acting Duke of End…

(Repetition of phrase, 3 paragraphs earlier.)

ooo

… roughly 89 percent… A distant second to this, at maybe 5 percent…

ooo

… Ghreni’s brain decided…

(A quirk with this writer, where a person and their brain isn’t the same thing.)

ooo

Coming in third, at maybe 4.5 percent…

ooo

Inasmuch as Blaine Turnin’s body…

(Repetition of full name.)

ooo

Coming in third, at maybe 4.5 percent of Ghreni’s cognitive attention, was I think I need a new minister of defense. Inasmuch as Blaine Turnin’s body was now presenting a shape that could only be described as “deeply pretzeled,” this was probably correct and therefore did not warrant any further contemplation.

(Repetition.)

ooo

And indeed, why Ghreni Nohamapetan?

ooo

And indeed, why Ghreni Nohamapetan? What were the circumstances of fate that led him to this moment of his life, spinning wildly out of control, literally and existentially, trying to keep from vomiting on the almost-certain corpse of his now-very-probably-erstwhile minister of defense?

ooo

That was, until a great shift in the Flow would happen-ed at some point in the near future…

ooo

(And I’m only 1/4 through this chapter.)

ooo

… rolling several times before coming to a full and complete stop.

ooo

Blaine Turnin’s body was in the seat opposite him, quiet, composed and restful, looking for all the world like he had not been a human maraca bean for the last half minute. Only Turnin’s head, tilted at an angle that suggested the bones in his neck had been replaced by overcooked pasta, suggested that he might not, in fact, be taking a small and entirely refreshing nap.

ooo

… in a secured room of his palace that lay far underground, in a subterranean wing…

ooo

I have to assume there are traitors in our my midst.

ooo

“Well, and this is just a hypothesis, it might have something to do with the fact that you’re an incompetent who assassinated his way to the dukedom and has lied to his subjects about the imminent collapse of civilization, which, incidentally, you have to date done nothing to prepare for in any meaningful way.”

ooo

“Why do you come to see me, Ghreni?” he asked.
“What do you mean?”
“I mean, why do you come see me?…”

ooo

“Yes, of course, you’re correct, an entirely ineffective rebel leader managed to infiltrate your security detail, plant at least one traitor, learn your secret travel itinerary and send a missile directly into your aircar and no others. Sorry, I was confused about that.”

(Repetition.)

ooo

“I need someone to talk to,” Ghreni said, suddenly.
Jamies looked over toward the (acting) duke. “I beg your pardon?”
“You asked why I keep visiting you,” Ghreni said. “I need someone to talk to.”

ooo

“I still vote for the therapist.”
“You could still help me,” Ghreni said.

ooo

Halfway there. Oh, I think I’ll stop here.

So. What’s going on here?

• A little bit of bad research or something.
• A little bit of not knowing, that 7 syllable names need to be shortened when used a lot.
• A lot of “repetitions are fun, if the language is different”.

For the last bit, ehm, no.

ETA: Rachel Neumeier kindly agreed to take a look at the same text: Writing Analysis: John Scalzi.

This week the question title is: A Pi Day Puzzle

You are planning a picnic on the remote tropical island of 𝜋-land. The island’s shape is a perfect semi-disk with two beaches, as illustrated below: Semicircular Beach (along the northern semicircular edge of the disk) and Diametric Beach (along the southern diameter of the disk).

If you pick a random spot on 𝜋-land for your picnic, what is the probability that it will be closer to Diametric Beach than to Semicircular Beach? (Unlike the illustrative diagram above, assume the beaches have zero width.)

And for extra credit:

Suppose the island of 𝜋-land, as described above, has a radius of 1 mile. That is, Diametric Beach has a length of 2 miles.

Again, you are picking a random point on the island for a picnic. On average, what will be the expected shortest distance to shore?

Highlight to reveal (possibly incorrect) solution:

Figure. Desmos.

From a point p₁ within the semi-disk, the distance to the semicircle is actually the distance to the point p₂ on the semicircle, that lies on the line going through the center of the semicircle and p₁, as that line is perpendicular to the tangent going through p₂. (Figure.) The distance to the diameter is measured by simply going straight down.

Assuming that the radius of the semicircle is 1, the distance from center to p₂ is 1. The distance from center to p₁ = (x₁, y₁) is (x₁² + y₁²)^0.5. The distance between p₁ and p₂ is 1 – (x₁² + y₁²)^0.5.

The distance from p₁ to the diameter is simply y₁.

And what we’re looking for is the part of the semi-disk, where y₁ < 1 – (x₁² + y₁²)^0.5.

The area of the whole semicircle is π * 1² / 2 = π / 2.

The area of the purple area of the Desmos figure is… Hang on.

y = 1 – (x² + y²)^0.5 <=>

1 – y = (x² + y²)^0.5 <=>

(1 – y)² = x² + y² <=>

1 – 2y + y² = x² + y² <=>

1 – 2y = x² <=>

1 – x² = 2y <=>

(1 – x²) / 2 = y

So the area of the purple area is

∫ (1 – x²) / 2 dx, from -1 to 1 =

1/2 ∫ 1 – x² dx, from -1 to 1 =

1/2 [x – x³/3] from -1 to 1 =

1/2 ((1 – 1/3) – (-1 + 1/3)) =

1/2 (2/3 + 2/3) =

2/3

So the probability of aiming for the red area and hitting the purple area is 2/3 / π/2 = 4/3π ~ 0.4244.

And for extra credit:

Program.

Imagine the semi-disk is the base of a shape, described by z = f(x, y). At the beaches, f(x, y) = 0. Elsewhere, f is the distance to the nearest shore. We want to integrate over this shape.

Or maybe we just want to monte carlo it. 😉

My program reaches the conclusion, that the average distance is something like 0.192 miles.

This week the question is: Can You Tip the Dominoes?

You are placing many, many dominoes in a straight line, one at a time. However, each time you place a domino, there is a 1 percent chance that you accidentally tip it over, causing a chain reaction that tips over all dominoes you’ve placed. After a chain reaction, you start over again.

If you do this many, many times, what can you expect the median (note: not the average) number of dominoes placed when a chain reaction occurs (including the domino that causes the chain reaction)? More precisely, if this median number is M, then you would expect to have placed fewer than M dominoes at most half the time, and more than M dominoes at most half the time.

And for extra credit:

You’re placing dominoes again, but this time the probability of knocking each domino over and causing a chain reaction isn’t 1/100, but rather 10^–k, where k is a whole number. When k = 1, the probability of knocking over a domino is 10 percent; when k = 2, this probability is 1 percent; when k = 3, this probability is 0.1 percent, and so on.

Suppose the expected median number of dominoes placed that initiates a chain reaction is M. As k gets very, very large, what value does M/10^k approach?

Highlight to reveal (possibly incorrect) solution:

Program 1, 2.

Calculating using probabilities: Assuming I simply start all these lines at the same time. After placing 1 domino in each line, 1/100 of them will tip over. After placing the 2nd domino in each remaining line, 1/100 of these will tip over. And so on, until more than half have tipped over. I write a short program to keep track of the numbers. The answer is 69.

Simulation: I write a short program to simulate a lot of lines. Same answer.

And for extra credit:

Program.

I change to first program a little, so I can go through probability 1/10, 1/100, 1/1000 etc. Apparently M/10^k goes towards 0.69314718, which looks like ln(2).

It is time to rant some! The internet doesn’t contain enough ranting!

Oscars

Sigh. Why do you do In Memoriam wrong every time? Or are the names, the pictures etc. only intended for the audience in the room?

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Indlæg på Facebook og kommentarer

Mere suk.

Jeg siger: “Det ville klæde pov at få en balance i de her indlæg, så vi ikke kun (næsten?) hører fra den ene side.”

Og så bliver jeg beskyldt for cancel culture, og jeg kan jo bare selv skrive et indlæg. At fortalere for transpersoners rettigheder er lige så gode som dem, der tror, Jorden er flad.

Bare suk.

At rejse er at være forvirret

Jeg har lige været ude at flyve med Norwegian. På nogle måder en interessant oplevelse. Først fik jeg fra anden side noget, der udtrykkeligt sagde, at det ikke var en billet. Hos Norwegian selv kunne jeg så hente et såkaldt rejsedokument. Det ligner en billet? Tæt nok på rejsen checkede jeg ind online. Og det var så her, jeg skulle regne ud, at hvis jeg hentede rejsedokumentetet igen, så havde det ændret sig og indeholdt et boarding pass. Ikke intuitivt.

En eller anden reklame

Der vist prøver at være morsom. Den taler i hvert fald om “bæredygtighed der holder”. Så … Bæredygtighed der er bæredygtig? Ha ha?

Politiken om ordet stealthing

I forbifarten skal det lige nævnes, at steak ikke udtales stiiik. Hvilket desværre er forkert. Jeg har oplevet fænomenet på Lolland. Og ordbogen godkender det også.

Oscar

De danske værter har ikke helt styr på at udtale Dolby.

This week the question is: How Many Rabbits Can You Pull out of a Hat?

I have a hat with six small toy rabbits: two are orange, two are green, and two purple. I shuffle the rabbits around and randomly draw them out one at a time without replacement (i.e., once I draw a rabbit out, I never put it back in again).

Your job is to guess the color of each rabbit I draw out. For each guess, you know the history of the rabbits I’ve already drawn. So if we’re down to the final rabbit in the hat, you should be able to predict its color with certainty.

Every time you correctly predict the color of the rabbit I draw, you earn a point. If you play optimally (i.e., to maximize how many points you get), how many points can you expect to earn on average?

And for extra credit:

Now, instead of two rabbits of each of the three colors, my hat contains 10. That is, it contains 10 orange rabbits, 10 green rabbits, and 10 purple rabbits. As before, every time you correctly predict the color of the rabbit I draw, you earn a point.

With optimal play, how many points can you expect to earn on average?

This was the 2nd attempt. Scroll down to find the weird 1st attempt.

Program. Figure 1, 2, 3.

If there’s only 1 rabbit left, I guess it correctly and get 1 point.

If there’s more than 1 rabbit left, I probably have to look at some cases. My guess has to be a color with the most rabbits left. The situation a orange rabbits, b green rabbits and c purple rabbits are already out AND the points related to that situation is denoted as [a][b][c]. Let’s look at the example [0][0][1].

• There are 5 rabbits left.
• My guess will be for orange, as that is a color with 2 rabbits left, the most possible.
• There’s p = 2/5 that the rabbit is actually orange.
• In that situation I get 1 point now and [1][0][1] later.
• There’s p = 2/5 that the rabbit is green.
• In that situation I get 0 points now and [0][1][1] later.
• There’s p = 1/5 that the rabbit is purple.
• In that situation I get 0 points now and [0][0][2] later.
• This means my expected points for this situation is
  • 2/5 * (1 + [1][0][1]) +
  • 2/5 * (0 + [0][1][1]) +
  • 1/5 * (0 + [0][0][2])
• I can’t calculate this sum until I have all the numbers.

This means it’s time for recursion! I build a program to handle all this, and get the result 3.3666666666667. I also do the calculation by hand and get 3 11/30.

And for extra credit:

I change a parameter in the program and run it again. The result is 13.703132371085.

1st attempt.

Program.

I’m not quite sure how I figured this. But I sort of said, that I could choose the rabbit color that gave me most points. For the fiddler, I actually got the correct number! But for extra credit, my number was too high, 15.367336492712. Well. I was on vacation. Too little time, too little brain power.

This week the question is: Can You Defend Your Trivia Knowledge?

A new season of LearnedLeague recently kicked off! Many folks may not be familiar with this daily trivia platform. I learned about it a few years ago and joined after my turn as a game show contestant.

Anyway, here’s how it works: Every day, you and your opponent for the day are presented with the same six trivia questions. You each do your best to answer these, and you assign point values for your opponent, without knowing (until the following day) which questions your opponent answered correctly. You must assign point values of 0, 1, 1, 2, 2, and 3 to the six questions.

For example, suppose I assigned values of 1, 2, 0, 3, 2, and 1 to the six questions, in order. Unbeknownst to me, my opponent answers the first, third, and fourth questions correctly. That means they get 1 + 0 + 3, or 4 points for the match. My own score depends on which questions I got right, and how these were scored by my opponent. If I get more than 4 points, I win the match.

Now, when someone answers three questions correctly, like my opponent just hypothetically did, the fewest points they can earn is 0 + 1 + 1 = 2, while the most points they can earn is 2 + 2 + 3 = 7. The fact that they got 4 points wasn’t great (from my perspective), but wasn’t terrible. In LearnedLeague, my defensive efficiency is defined as the maximum possible points allowed minus actual points allowed, divided by the maximum possible points allowed minus the minimum possible points allowed. Here, that was (7−4)/(7−2), which simplified to 3/5, or 60 percent.

By this definition, defensive efficiency ranges somewhere between 0 and 100 percent. (That is, assuming it’s even defined—which it’s not when your opponent gets zero questions right or all six questions right.)

Suppose you know for a fact that your opponent will get two questions right. However, you have absolutely no idea which two questions these are, and so you randomly apply the six point values to the six questions.

What is the probability that your defensive efficiency for the day will be greater than 50 percent?

And for extra credit:

Now suppose your opponent is equally likely to get one, two, three, four, or five questions correct.

As before, you randomly apply the six point values (0, 1, 1, 2, 2, 3) to the six questions.

What is the probability that your defensive efficiency will be greater than 50 percent?

Highlight to reveal (possibly incorrect) solution:

Program.

6 questions. Points are 0, 1, 1, 2, 2, 3. My opponent had 2 correct. What is the probability my DE > 50%?

• The minimum possible points is 0 + 1 = 1.
• The maximum possible points is 3 + 2 = 5.
• I want DE > 0.5. (1)
• DE = (5 – actual) / (5 – 1)
• DE = (5 – actual) / 4 (2)
• Combine (1) and (2)
• (5 – actual) / 4 > 0.5
• 5 – actual > 2
• 3 > actual

These are the possible point values, point combinations and number of permutations for my opponent:

Actual	Combo	Perms	Combo	Perms	Total
1	0 + 1	2			2
2	0 + 2	2	1 + 1	1	3
3	0 + 3	1	1 + 2	4	5
4	1 + 3	2	2 + 2	1	3
5	2 + 3	2			2
					15

Of these row 1 and 2 are good, because actual < 3. There are 5 good permutations out of 15 in all. So the probability I’m looking for is 5/15 = 1/3. And my program agrees.

And for extra credit.

Program.

To get this one, I just have to calculate all the probabilities for 1-5 correct guesses and average them out.

Actual	Probability
1	1/2
2	1/3
3	1/2
4	1/3
5	1/2
1-5	13/30

The looked for probability is 13/30 = 0.43333… My program agrees.

It is time to rant some! The internet doesn’t contain enough ranting!

Hark! Two Holiday Puzzles

“For strings of length seven or longer (e.g. a phone number or a Social Security number) it appears you can always find an equation — with a single exception.”

An American phone number, love.

372 Pages We’ll Never Get Back, episode 181

1) Rowling (the Harry Potter one) rhymes with rolling. 2) Go look at my statistics spreadsheet!

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Vestens politiske korrekthed koster kvindeliv og -rettigheder

Politiet er ikke altid lige optaget af at opklare voldtægter. Ja, det er skidt.

Lovgivningen taler om personer med en livmoder og sådan noget. Og det er på en eller anden måde lige så skidt, ja, livsfarligt? Jeez.

København har fundet på, at der i februar skal være lysfestival. Så mellem 17 og 22 kan man beskue kunstværker med lys i, udendørs, ganske gratis. Egentlig slet ikke så dumt. ✅

Da nogle af værkerne befinder sig et stykke fra centrum, kan man deltage i en guidet tur, ombord på en bus. Rigtig god idé. ✅

Guiderne er også arrangørerne. De kender værkerne og sådan noget og kan små anekdoter. Genialt. ✅

Arrangørerne skaffer også finansiering til hele apparatet, så de er smaddergode til det sprog. Det fremhæves, at den tilknyttede app er noget helt særligt. Vi finder ud af, hvordan den her festival adskiller sig fra dem i resten af verden. Forbløffende mange ting er gode, sjove, fantastiske osv.

Yrk. ❌

En guidet tur skal lære mig noget om seværdighederne. Det her værk med lys i, fortæl mig alt! Og hvis der er tid tilovers, så må du også godt sige andre ting. Fortæl mig til gengæld ikke, at alting er godt, det skal jeg nok selv tage stilling til. ❌

Det er en kunst at være en god guide. Har vi alle med? Kan alle høre? Til tider bliver der i bussen berettet om det, vi kommer til om lidt. Og så er der 10 minutter til at se tingene, og så kører vi, og hvis du ikke kan finde ud af det, så tag s-toget hjem. ❌

Og lad være med at trække os ind i mørket uden lommelygter, på glat grund. ❌

Hvis du ikke er guide, så find en. ❌

This week the question is: Can You Squeeze the Heart?

You can generate a heart shape by drawing a unit square (i.e., a square with side length 1), and then attaching semicircles (each with radius 1/2) to adjacent edges, as shown in the diagram below:

What is the radius of the smallest circle that contains this heart shape?

And for extra credit:

Instead of containing one heart shape, now your circle must contain two heart shapes. Again, each heart consists of a unit square and two semicircular lobes. The two hearts are not allowed to overlap.

What is the radius of the smallest circle that contains these two hearts?

Before moving on to the solutions. Last week I did this:

• Sent in a solution.
• Realized that while the method was right, the numbers ended up being wrong.
• Postponed sending in a new solution.
• Forgot!
• Felt very sorry for myself.

Sigh.

Highlight to reveal (possibly incorrect) solution:

Desmos.

The circumscribed circle passes through the lowest tip of the square. The center of the circle is above that point. I create a simulation in Desmos of the circle, using these 2 items of information. I introduce a variable a for the radius of this circle. I then let a wobble between too small and too large, to get an estimate of a. Zoom into one of the points of the circle, where the wobble is visible, where the circle passes through the tangent point. Repeat. I kept going until I had a between 0.891805 and 0.891806.

And for extra credit:

Figure 1, 2, 3, 4. Desmos 1, 2, 3.

I tried a number on variations on this.

When the 2 hearts point in opposite directions, I can slide them against each other. One configuration is that the 2 squares are neatly lined up, figure 1. Then the radius is a = 1.5. If I slide them “away from” each other, figure 2, a grows bigger. If I slide them “closer to” each other, figure 3, a grows bigger. So if this is indeed the way they are arranged, a = 1.5.

Another option is that the hearts are sort of tucked into each other, figure 4. But again, then a has to be bigger.

The last Desmos attempt looks at one heart right way up, the other tilted 45°. No joy.

There might be other configurations I just can’t think of. Sigh.

Recently I had to reread ” #Hell Is the Absence of #God “, #TedChiang . Because it’s good. And I became fascinated with the #worldbuilding . So, let me tell you the rules of this world. At least as they appear to the people in the story. Beware spoilers.

• God exists. (1)
  • As with the rest of these items, faith isn’t involved in the system, everybody knows how it works.
  • God isn’t involved in everything. (1)
• Heaven (up) exists. (5)
  • Heaven is good. (8)
• Hell (down) exists. (7)
  • Hell is exile from God. (7)
• There is an afterlife.
  • Those who love God perfectly go up, including successful light-seekers (see below). (5, 9, 23, 24)
  • Suicides go down. (14)
  • Failed light-seekers (dying in the attempt) go down. (26)
  • Humanists go down. (16)
  • The dead in Heaven can visit Earth. (11,18) They don’t talk, but they seem happy. (11) They don’t have bodies. (25)
  • The dead in Hell can be observed by the living, but not the other way around. (7) They seem mentally content and physically fine. (7, 15)
• Angels exist. (3)
  • From time to time, angels visit Earth. (1) The timing is random. (10, 26) Some sites (all inhospitable) are visited often. (13, 22, 27)
  • A visit from an angel has a lot of impact: miracles including cures (3, 10), and consequences of the violent entrance and exit (4), including instant knowledge of the existence of God (becoming eyeless) (3, 19, 23), and injury and death (6). All impact is random. (3, 12, 20, 21)
  • Some angels have fallen, their impact seem less violent, and it’s possible to talk to them. (17)
• Light-seekers seek out angels to become eyeless. (24)

So, those are the rules. On top of that, people try to interpret the visitations etc. They don’t appear to have much success with that (because of the randomness), so that’s a separate part of the world building, and I won’t look at it here.

I bolded 2 parts of the rules above, because the story illustrates, that 2 rules can be in conflict. (28) Leading to Hell feeling unbearable for a few. (29)

For completeness, here are the quotes in the order they appear in the text.

1. Neil was born with a congenital abnormality that caused his left thigh to be externally rotated and several inches shorter than his right; the medical term for it was proximal femoral focus deficiency. Most people he met assumed God was responsible for this, but Neil’s mother hadn’t witnessed any visitations while carrying him; his condition was the result of improper limb development during the sixth week of gestation, nothing more. In fact, as far as Neil’s mother was concerned, blame rested with his absent father, whose income might have made corrective surgery a possibility, although she never expressed this sentiment aloud.
2. He became an adult who — like so many others — viewed God’s actions in the abstract until they impinged upon his own life. Angelic visitations were events that befell other people, reaching him only via reports on the nightly news.
3. It was an unexceptional visitation, smaller in magnitude than most but no different in kind, bringing blessings to some and disaster to others. In this instance the angel was Nathanael, making an appearance in a downtown shopping district. Four miracle cures were effected: the elimination of carcinomas in two individuals, the regeneration of the spinal cord in a paraplegic, and the restoration of sight to a recently blinded person. There were also two miracles that were not cures: a delivery van, whose driver had fainted at the sight of the angel, was halted before it could overrun a busy sidewalk; another man was caught in a shaft of Heaven’s light when the angel departed, erasing his eyes but ensuring his devotion.
4. […] the angel’s billowing curtain of flame […]
5. […] her soul’s ascension toward Heaven.
6. Nathanael hadn’t delivered any specific message; the angel’s parting words, which had boomed out across the entire visitation site, were the typical Behold the power of the Lord. Of the eight casualties that day, three souls were accepted into Heaven and five were not, a closer ratio than the average for deaths by all causes.
7. Like every other nondevout person, Neil had never expended much energy on where his soul would end up; he’d always assumed his destination was Hell, and he accepted that. That was the way of things, and Hell, after all, was not physically worse than the mortal plane. It meant permanent exile from God, no more and no less; the truth of this was plain for anyone to see on those occasions when Hell manifested itself. These happened on a regular basis; the ground seemed to become transparent, and you could see Hell as if you were looking through a hole in the floor. The lost souls looked no different than the living, their eternal bodies resembling mortal ones. You couldn’t communicate with them — their exile from God meant that they couldn’t apprehend the mortal plane where His actions were still felt — but as long as the manifestation lasted you could hear them talk, laugh, or cry, just as they had when they were alive.
8. Of course, everyone knew that Heaven was incomparably superior […]
9. Now that Sarah was in Heaven, his situation had changed. Neil wanted more than anything to be reunited with her, and the only way to get to Heaven was to love God with all his heart.
10. When Janice’s mother was eight months pregnant with her, she lost control of the car she was driving and collided with a telephone pole during a sudden hailstorm, fists of ice dropping out of a clear blue sky and littering the road like a spill of giant ball bearings. She was sitting in her car, shaken but unhurt, when she saw a knot of silver flames — later identified as the angel Bardiel — float across the sky. The sight petrified her, but not so much that she didn’t notice the peculiar settling sensation in her womb. A subsequent ultrasound revealed that the unborn Janice Reilly no longer had legs; flipperlike feet grew directly from her hip sockets.
11. Janice’s parents were sitting at their kitchen table, crying and asking what they had done to deserve this, when they received a vision: the saved souls of four deceased relatives appeared before them, suffusing the kitchen with a golden glow. The saved never spoke, but their beatific smiles induced a feeling of serenity in whoever saw them.
12. There Janice met two individuals with cancer who’d witnessed Rashiel’s visitation, thought their cure was at hand, and been bitterly disappointed when they realized they’d been passed over.
13. […] the holy sites, those places where — for reasons unknown — angelic visitations occurred on a regular basis […]
14. If suicide would have ended his pain, he’d have done it without hesitation, but that would only ensure that his separation from Sarah was permanent.
15. […] she’d seen her husband among the lost souls. […] she had committed suicide to rejoin her husband. None of them knew the status of Robin’s and her husband’s relationship in the afterlife, but successes were known to happen; some couples had indeed been happily reunited through suicide.
16. […] the humanist movement; its members considered it wrong to love a God who inflicted such pain, and advocated that people act according to their own moral sense instead of being guided by the carrot and the stick. These were people who, when they died, descended to Hell in proud defiance of God.
17. Visitations of fallen angels were infrequent, and caused neither good fortune nor bad; they weren’t acting under God’s direction, but just passing through the mortal plane as they went about their unimaginable business. On the occasions they appeared, people would ask them questions: Did they know God’s intentions? Why had they rebelled? The fallen angels’ reply was always the same: Decide for yourselves. That is what we did. We advise you to do the same.
18. […] visions don’t appear just because a person needs one […]
19. […] few visitations resulted in an eyeless person, since Heaven’s light entered the mortal plane only in the brief moments that an angel emerged from or reentered Heaven […] The light that had brought his soul as close to perfection as was possible in the mortal plane had also deformed his body […] Benny described Heaven’s light as infinitely beautiful, a sight of such compelling majesty that it vanquished all doubts. It constituted incontrovertible proof that God should be loved, an explanation that made it as obvious as 1+1=2. [More about Benny below, at 23.]
20. Neither of them had ever heard of a previous instance where God had left His mark on a person in one visitation and removed it in another.
21. There were a few instances of individuals receiving multiple miracle cures over their lifetimes, but their illnesses or disabilities had always been of natural origin, not given to them in a visitation.
22. Whereas in most of the world one could wait an entire lifetime and never experience a visitation, at a holy site one might only wait months, sometimes weeks. Pilgrims knew that the odds of being cured were still poor; of those who stayed long enough to witness a visitation, the majority did not receive a cure.
23. […] the absoluteness of Benny’s devotion. No matter what misfortune befell him in the future, Benny’s love of God would never waver, and he would ascend to Heaven when he died.
24. Every holy site had its pilgrims who, rather than looking for a ­miracle cure, deliberately sought out Heaven’s light. Those who saw it were always accepted into Heaven when they died, no matter how selfish their motives had been […]
25. [Becoming eyeless:] At an instinctual level, Neil was averse to the idea: it sounded like undergoing brainwashing as a cure for depression. He couldn’t help but think that it would change his personality so drastically that he’d cease to be himself. Then he remembered that everyone in Heaven had undergone a similar transformation; the saved were just like the eyeless except that they no longer had bodies.
26. […] seeking Heaven’s light was far more difficult than an ordinary pilgrimage, and far more dangerous. Heaven’s light leaked through only when an angel entered or left the mortal plane, and since there was no way to predict where an angel would first appear, light-seekers had to converge on the angel after its arrival and follow it until its departure. To maximize their chances of being in the narrow shaft of Heaven’s light, they followed the angel as closely as possible during its visitation; depending on the angel involved, this might mean staying alongside the funnel of a tornado, the wavefront of a flash flood, or the expanding tip of a chasm as it split apart the landscape. Far more light-seekers died in the attempt than succeeded. Statistics about the souls of failed light-seekers were difficult to compile, since there were few witnesses to such expeditions, but the numbers so far were not encouraging. In sharp contrast to ordinary pilgrims who died without receiving their sought-after cure, of which roughly half were admitted into Heaven, every single failed light-seeker had descended to Hell.
27. Holy sites were invariably in inhospitable places: one was an atoll in the middle of the ocean, while another was in the mountains at an elevation of twenty thousand feet.
28. Neil began, slowly but surely, bleeding to death. […] another beam of Heaven’s light penetrated the cloud cover and struck Neil […] the light revealed to Neil all the reasons he should love God. […] So minutes later, when Neil finally bled to death, he was truly worthy of salvation. And God sent him to Hell anyway. [Ethan] saw Neil’s soul leave his body and rise toward Heaven, only to descend into Hell.
29. Everything Neil sees, hears, or touches causes him distress, and unlike in the mortal plane this pain is not a form of God’s love, but a consequence of His absence.