Dear 100 Hour Board,

So. I read about something Called the Chinese Soldier Theorem in algebra that made me think the following questions (which I think about every time I do a session at the Salt Lake Temple) might actually be solvable.

Premise 1: In the Salt Lake Temple when you do endowments, you move from room to room, for a total of five rooms before you get to the Celestial Room, counting the chapel where you wait.

Premise 2: As you move from room to room, the order starts at the head of each row until the row is finished and then restarts at the head of the next row.

Premise 3: Even if you were technically seated in one of the front rows in the chapel, the seating format shuffles people up to the extent that, by the time the session is finished, someone who was seated in the front row in the chapel could be seated in the middle or back rows in the Telestial Room.

Premise 4: Seat #1 is the seat on both sides used by the man and woman acting as the witness couple.

Questions:

1. For each room in the Salt Lake Temple, how many seats are in each row in each room on the female side, starting in the Creation Room?

2. Assuming that there are six rows of women starting in the Creation Room and that no seat is ever empty, how many seats can you sit from seat #1 and still be guaranteed that you will always be sitting in the first row in every room?

2.5 Assuming the same as in #2, how many seats can you sit from seat #1 and still be guaranteed a seat in the second row in every room?

3. Assuming the same as #2, if the temple matrons leave an empty seat on the aisle seat of three of the six rows starting in the Garden room, how many seats can you sit from seat #1 and still be guaranteed a seat in the second row of every room?

4. Assuming that there were six rows of women to start with in the Creation Room and that the temple matrons created three empty aisle seats starting in the Garden Room and continuing into the other two rooms, and assuming that no row in the Terrestrial Room was seated with more people than the second row from the front, how many people would be seated in each row, and would there still be six rows of people (or more or less)? How many seats would you have to sit from seat #1, starting in the Creation Room, to still be seated within the first two rows in the Terrestrial Room?

Thanks!

-Sand Dollar

Dear Reader,

Sorry this took a bit longer than I promised in Board Question #89327. While I was able to go the temple as planned, I got slammed with finals and a subsequent case of burned-out-itis, causing this to be held up longer than I originally thought. But without further ado, let's get around to actually answering you. (Now even more time has passed since I originally penned this paragraph, meriting another opening/apology.)

I come to you midst broken promises of times when I would have this finished. And, quite honestly, personal broken dreams for when I would do this. When I initially saw your question appear in the inbox, I immediately was drawn to it simply because it mentioned the word "theorem". With eerie prescience, I set forth a placeholder that said "I feel an obligation to answer this just cuz the word theorem was mentioned. It will probably go waaaay over hours, though." Little did I know just how true that statement would turn out to be.

But no more of my apologetic/pathetic rambling; I humbly proffer you the best answer I was able to cobble together:

To start off, assuming what I found is the same thing you were referencing in your question, the Chinese Remainder Theorem is actually a part of number theory, specifically dealing with modular functions. While it does have some pretty cool applications (like in RSA encryption), it doesn't have as many implications in optimization, which is what you want. So while I thank you for the interesting read on number theory, I'm going to go the optimization route to provide an answer (*Note that this is completely different from dynamic optimization as well--thank goodness, cuz as hard as I'm trying to learn Optimal Control Theory in order to do my job, a large portion of it still goes over my head). Haha, jk, jk, turns out I never really figured this out either. But never fear; your answer still awaits below!

The first step to answering was of course going to the SLC Temple to gather data. Auto Surf and Kirito were nice enough to accompany me, and it was with Auto's help that I counted the rows and columns. Quick disclaimer here: I had no writing utensils on me, and so I just memorized the numbers for rows and columns for each room, and then sent myself an email with those numbers afterwards. However, not anticipating the drag between going to the temple and writing up my response, that email solely consisted of "794, 8910, 912, 9105". So, while I am fairly certain I remember whether each of those numbers represented rows or columns, there might be some error stemming from that. Also note that I'm attempting to solve this with the calculator on my phone whilst holed up in the library (so probably take my findings with a grain of salt). (True to the form of my life, though I wrote that previous sentence over a month ago, it still perfectly describes my exact situation in presently attempting to solve this.)

**1. For each room in the Salt Lake Temple, how many seats are in each row in each room on the female side, starting in the Creation Room?**

Creation Room: Okay, this room has a pretty crazy arrangement of seats, which reminded me of a seashell overall. There are five rows at the front of the room, positioned in a sort of curvy diagonal, with 7 seats in the first row, 9 seats for rows 2-4, and 4 seats in the back row. Then there's a more block like group of seats at the back of the room, which is 7 rows, 9 columns (or... that might be flipped). Totaling these two sections together yields 101 seats for the female side.

Garden Room: I have to admit that I took a breath of relief when I saw I wouldn't have to memorize as strange an assortment of seats here. The arrangement pretty much sticks to a rectangle, just with less seats in the first row than the others. There are 9 rows overall, with 8 seats in the first, and 10 seats in all the others. So the seat total comes to 88 (honestly, this number makes me just a bit dubious because it's so far off from the totals for the other rooms, but oh well).

In Between Room (thusly dubbed because apart from the seats, all I remember about it is that it came in between the Garden Room and the Terrestrial Room): This was the nicest room to count seats in. Everything was a perfect rectangle! (I never knew just how much I value rectangles until having to count and remember random numbers for a few hours.) Anyways, here there are 12 rows with 9 columns, coming to a nice total of 108 seats. (Edit: after re-reading your question, I realized this must be the Telestial Room. But hey, the name still works, right? After all, we live in a Telestial world, and our time here is our in between state of not being mortal... )

Terrestrial Room: And we're back to different numbers of seats per row. At the back of the room, there was a cluster of rows with less seats than normal, but unfortunately, as I was--slowly, cause apparently counting and moving is above my multi-tasking skills--walking up, I didn't get the chance to count just how many of those rows or seats per row there were. Excluding that bunch at the back, in the foremost row, there were 5 seats, and then 9 seats per row for the following 9 rows. Adding those up gives 86 seats total (considering this is pretty similar to the Garden Room, I'm thinking there might have been a comparable section of seats I missed in there as well.

Overall, it seems as though there were about 100 seats per room on the female side leading up to the Celestial Room.

**2. Assuming that there are six rows of women starting in the Creation Room and that no seat is ever empty, how many seats can you sit from seat #1 and still be guaranteed that you will always be sitting in the first row in every room?**

And this is where the fun math part kicks in. As previously mentioned, I didn't ever get struck over the head by some beneficent math fairy of optimization, so instead of using half-baked formulas, I'm going to go painstakingly old school with the aid of the following charts of the layouts of each room I drew up in a notebook:

With the use of these handy-dandy diagrams, finding your answer now becomes a matter of testing out all the different seats to see which allows you to always be in the first row. Thanks to the Terrestrial Room, we automatically know it can't be more than 5. Now, in one of your premises, you state that the order follows the heads of each row. However, because the witness woman (I don't really know what else to call the woman who's part of the witness couple... ), always has to sit in the leftmost seat, I'm going to say the "head" of the row becomes the person sitting directly on the right of the witness woman.

Numbering all the seats in my pictures (numbering left to right according to seating in the Garden Room) reveals that people 1-4 are always guaranteed a spot in row 1. It also reveals that person 18 will be sitting in the front row in the Terrestrial Room, while poor person 19 is stuck in the middle of the 5th row (who would have thought).

**3. Assuming the same as in #2, how many seats can you sit from seat #1 and still be guaranteed a seat in the second row in every room?**

Since you haven't added conditions on the aisle seats being empty yet, I can use the same numbering as for #2. A quick look tells me that lucky #13 is the highest secured second row seat (which is the 4th seat from the right on the second row in the Creation Room).

**4. Assuming the same as #2, if the temple matrons leave an empty seat on the aisle seat of three of the six rows starting in the Garden room, how many seats can you sit from seat #1 and still be guaranteed a seat in the second row of every room?**

Again, the witness woman requires a little specification with your premises, so I'm going to leave an empty aisle seat starting on the 2nd row. This time 12 is the highest seat number to stay in the 2nd row. As a bonus, it's still 1-4 that are always in row 1, though this time 17 is the random higher seat in the front in the Terrestrial Room.

**5. Assuming that there were six rows of women to start with in the Creation Room and that the temple matrons created three empty aisle seats starting in the Garden Room and continuing into the other two rooms, and assuming that no row in the Terrestrial Room was seated with more people than the second row from the front, how many people would be seated in each row, and would there still be six rows of people (or more or less)? How many seats would you have to sit from seat #1, starting in the Creation Room, to still be seated within the first two rows in the Terrestrial Room? **

Okay, This question needs the most accuracy disclaimers attached to it, because I'm not even totally certain that I'm interpreting it exactly correctly (so all y'all know what I'm doing, I'm just leaving three seats empty per row starting in the 2nd row in the Garden Room). Nevertheless, I shall endeavor to do my best.

To make sure I was doing this right, I copied my charts in order to re-number everything. Thanks to my tiny handwriting, pictures I took of my work with my phone are nigh illegible, and I don't particularly want to attach 4 pictures of badly drawn squares and lines with minuscule numbers hovering over the top. If anyone would like to see where all the people end up, shoot me an email, and I'll be more than happy to give you a copy of my results.

The final results turned up 8 rows in the Terrestrial Room. This time the seat which started out in the second row in the Creation room and ended up on the front row in the Terrestrial Room was 15. However, 11-14 were all in the third row of the Terrestrial Room. The highest seat number where all the seats below it are also guaranteed being within the first two rows was 10. Somehow 23 got pushed back from the third row in the Creation Room to the 6th row in the Terrestrial Room. 47 actually made some decent progress, moving up in the world an entire row. And the very last person to go through the veil would be 37.

So, there you have it. Hopefully it was worth at least part of the wait.

In closing, I'd just like to leave with a scripture which I have found the seating in the Salt Lake Temple to cast considerable light upon:

~Anathema

Dear friend,

Thanks for getting me to the Salt Lake Temple. It had been too long.

Also, pro tip: Go with people smarter than you and listen to what they learn.

Pro-er tip: Realize everyone is smarter than you in some way.

Pro-er-er tip: Invest in real photo editing software.

Take care,

-Auto Surf

Dear Cinnamon,

I actually go to a session at the Salt Lake temple often, and questions like this plague me (and my friends) all the time. I actually almost don't want to go this week because the woman who chooses the seating arrangements for the women's side has been doing it WRONG the past couple weeks, so I keep getting pushed back rows when I feel it's unnecessary. (It's unlikely anyone would run into me at the temple, because I'm not saying when my usual time is, but on the off-chance you hear someone mutter "Unsustainable" as we're seated in the Telestial room, congrats, you just heard a writer.)

With the knowledge born from years of observation, off the top of my head you have to be within five seats of #1 to stay on the first row the whole session and either 11 or 14 away from #1 to be guaranteed the second row (they almost never fill the second row in the Terrestrial room because of visibility issues, which is whence the lower number comes).

So, anyway, basically what the kids said above (they actually tried a lot harder than I did, because they used their brains and I just used the fire of my rage).

Love,

-MSJ