Today is a very special day in the history of our country. That's right -- it's Palindrome Day and the first day of another Palindrome Week.
First of all, it's Palindrome Day according to the m-dd-yyyy format, since it is 1-20-2021. But if we write it in the m-dd-yy format, then it is 1-20-21, which the first day of a "week" of palindromes. The current decade is filled with such Palindrome Days and Weeks -- the next Palindrome Week is in February 2022, with the first day of that week, the 20th, a Palindrome Day.
Today I subbed in a high school special ed class. What makes it significant is that it's my first day subbing in my first Orange County district since before the long-term job in my new district (which was in fact just before hybrid began). And so it's worth it to note the differences between my two districts.
First of all, the regular teacher has a plexiglass shield around her desk, and the students also have smaller glass trifolds for their desks. Also, the thermometer is mounted near the front door. And there is an air purifier near the back of the room. My new district has none of these safeguards -- and we had to operate the thermometers there by hand.
As for the bell schedule, I've already stated that the order of classes at high schools are the same in both districts --Cohort A meets Tuesday (odd periods) and Wednesday (evens) while Cohort B meets Thursday (odds) and Fridays (evens). In both districts now, students must log in to Zoom on the days their cohort doesn't meet face-to-face. But in today's district, each class is 55 minutes and so all three classes are completed by lunch -- an optional tutorial is after lunch for students who need it.
I've subbed in this classroom before -- the last time I subbed here was, interestingly enough, also during a Palindrome Week (9-11-19). Today, the second period class is senior Economics, fourth period is World History, and sixth is her conference period. Back in September 2019, I subbed for her Business Math class as well, but that was before the coronavirus and the block schedule. This year, her Business Math meets during an odd period.
In Econ, the students work on Quizizz to prepare for their Chapter 1 Test -- Econ in California has always been a one-semester course, so this is only their third week here. The test covers the basics of Econ -- was is scarcity, what are goods and services, and so on. The World History class is studying European colonization of Africa. It has only one in-person student, a guy -- but then again, special ed classes like this were always small anyway. Indeed, the class even has an aide for special ed -- and so there are more adults in the class than students. (It goes without saying that there's no "A Day in the Life" today.)
When I was this room in September 2019, I didn't sing any songs -- not even the Palindrome Song. So today, I correct that omission and sing the Palindrome Song from Square One TV:
PALINDROME SONG
Refrain:
Go forward then go backward.
If the number reads the same,
Then it's a palindrome.
Go backward then go forward.
If the number's still the same,
Then it's a palindrome.
It's not a palomino,
On the western plain,
Where cowboys love to roam.
No! The wonder of all wonders,
A backward-forward number.
It's called a palindrome!
1st Verse:
Let's see! Take a 33!
Read it in reverse.
Hey it reads the same,
So it's a palindrome.
And 505,
Or 2002,
So they're both examples of that one I love.
It's a palindrome!
It's a palindrome!
It's a palindrome!
(repeat Refrain)
2nd Verse:
Let's see! Take a 63!
Read it in reverse.
Now we're in a fix,
'Cause it's a 36.
But don't be sad,
All we do is add.
Add the 63 to 36 and see.
That's a 99!
Hey we're doing fine!
That's a palindrome! Whoa! Whoa!
(repeat Refrain)
Actually, the refrain changes slightly each time it's sung. After the first verse, the palomino is replaced with a pachyderm. But I'm really having trouble understanding those lines completely:
Then it's a palindrome.
It's not a pachyderm,
Performing in the circus,
At the hippodrome.
No! The wonder of all wonders...
Hmm, a "pachyderm" is an elephant, and elephants do appear in a circus, but a "hippodrome" is an ancient horse track, not any place for elephants. Then again, sometimes the words "circus" and "hippodrome" were used interchangeably, such as the famous Hippodrome of Byzantium. (Or should I say, Constantinople?)
After the second verse, the following lines appear (starting in the middle at "Go backward"). These lines are more important, since they explain the process to generate a palindrome:
Go backward and go forward.
It's the one that's glad you came,
'Cause it's a palindrome.
If reversing and then adding,
Doesn't work the first time,
Repeat it 'til you're home.
'Cause you'll finally reach that number,
That backward-forward number,
You've reached a palindrome!*
(*unless, of course, you started with a Lychrel number like 196)
In 2019, I was in this room on Patriot Day, and yes, 9/11 attacks were mentioned in the lesson. This year, I visit the classroom not on Patriot Day, but on Inauguration Day --
-- yes, look at how far I get in this post without mentioning the inauguration. As is typical for the classroom, I do try to show the students the DC ceremony. The tricky thing is that at this school, the tardy bell for the earliest class rings at 9:00 -- that is, noon Eastern time. And since it's been months since I've subbed in this district (and the first time in this district for hybrid), I need some time and assistance to learn district policies and set up my Chromebook for Zoom. By the time I finally get everything set up and go to YouTube for the livestream, Joe Biden has already taken the oath of office and is getting ready to start his inauguration address.
(Had I still been at my long-term middle school, school would have started 8:45, leaving a cushion between the tardy bell and the inauguration. The first class would have been Math 7. But then there might have been confusion since this is the first week of logging in to Google Meet on off-days -- many Tuesday/Thursday kids would have been trickling in, possibly distracting me from the inauguration.)
One student (who remembers me from the last time I subbed at this school) does ask for my opinion of the new administration. I stay politically neutral by avoiding revealing which candidate I voted for. I do reply that I hope the incoming administration is successful, and that it is nice to have a Californian, Kamala Harris, as the Vice President. I also point out that the first inauguration I watched at school was that of George HW Bush, when I was a second grader.
Four years ago, I was at the old charter school during the inauguration of Donald Trump. I showed the ceremony to my sixth graders. Many of them asked me which candidate I voted for -- and I truthfully replied that I'd voted for neither, since my car unexpectedly broke down on Election Day.
Other inaugurations that took place when I was in school include Bill Clinton's first (as a young sixth grader) and Barack Obama's first (as a sub, just before I started student teaching). The only other inauguration on a weekday was George W Bush's second, but I wasn't in the classroom -- ironically, I was close to Washington DC just ten days after that inauguration, when I was applying for the NSA job.
Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
A trirectangular tetrahedron has edges of length 44, 117, 125, 240, 244, 267. If the legs are x, y, z, what is sqrt(x + y + z - 1)?
Actually, a "trirectangular tetrahedron" is one that is exactly half of a box. Three of its faces are right triangles and the fourth is a triangle whose sides are the hypotenuses of the other faces.
So three of the edges are also legs -- including the two shortest, 44 and 117. We can use trial-and-error and the Pythagorean Theorem to determine the third leg -- we find that 44^2 + 240^2 = 244^2, and so 240 is the third leg.
All that remains is to plug in 44, 117, 240 for x, y, z in the expression:
sqrt(44 + 117 + 240 - 1) = sqrt(401 - 1) = sqrt(400) = 20
Therefore the desired value is 20 -- and of course, today's date is the twentieth.
Lecture 6 of Prof. Arthur Benjamin's The Mathematics of Games and Puzzles: From Cards to Sudoku is called "Expert Backgammon." Here is a summary of the lecture:
- Harvey Gillis, the successful CEO of a venture capital firm, once wrote about an experience he once had playing Backgammon. He described it like a military conflict or risky business deal.
- In Backgammon, White starts with 2 checkers at point 1, 3 at 17, and 5 each at 12 and 19 and moves counterclockwise from 1 to 24. Black starts opposite and moves clockwise from 24 to 1.
- Each player must move all checkers into the home board (last 6 points) and then can bear them off the board. The first player to remove all checkers is the winner.
- To start, each player rolls 1 die and the higher roll goes first. If Black rolls 3 and White rolls 1, then Black goes first with the roll 3-1. He can move one checker 3 point and another 1 point.
- If a player has at least two checkers on a point, that point is "made" and the opponent can't place a checker there. Rolling doubles counts as rolling four of a kind. This occurs with probability 1/6.
- A lone checker on a point is a blot. If the opponent hits a blot, that checker is placed on the bar, and that opponent roll it back on the board before making any other move.
- Best opening rolls include 3-1 (for Black, making the 5 point), 4-2 (making the 4 point), 6-1 (making the 7 point), and 5-3 (making the 3 point).
- For 6-2, 6-3, 6-4, it's better for Black to come out from 24 to 18, and then advance that same checker from 18 or else advance one from 13. For 6-5, move directly from 24 to 13.
- For 5-2, it's best to advance both from 13, one to 8 and the other to 11. For 5-4, advance one from 13 to 8 and either 13 to 9 or 24 to 20. For 5-1, advance to 13 and go from 24 to 23.
- When both dice are small, advance one from 13 (midpoint) and the other from 24 (back). But for 2-1, one might consider advancing one from the midpoint and the other from 6 to 5.
- If Black opens with 5-4 and places a blot on 9, Black's blot is 8 pips away from White. White can hit this blot with an indirect shot involving both dice. It's almost always correct to hit blots.
- There are eleven rolls containing a 1, eleven rolls containing a 2, and so on up to 6. Thus there is at least 11/36 or a 30% of hitting a blot that's six or fewer pips away -- a direct shot.
- The "race" begins when each player is unlikely to hit the other, and so both players are getting ready to bear off or remove checkers. The pip count determines which player has the lead.
- If all checkers are borne off before opponent bears off any, it's gammon/double win. Plus if it's an opponent checker left in home board, it's backgammon/triple win. There's also a doubling cube.
- If you have at least a 25% chance of winning, you should take the double. Here's why -- if you fold you'll always lose 1, but if you take you either gain 2 or lose 2, so it's getting 3:1 pot odds.
- In 2-roll position (both White and Black have four checkers on last, Ace point) then White should take Black's double. It's likewise for 3-roll position, but for 4-roll position White should take.
Lesson 9-2 of the U of Chicago text is called "Prisms and Cylinders." Our text refers to both prisms and cylinders as "cylindric surfaces."
This is what I wrote last year about today's lesson:
This lesson consists mainly of definitions. Terms defined in this lesson are surface, solid, box, rectangular solid, faces, opposite faces, edges, vertices, and skew lines -- and that's just the first page! Another term defined in this lesson is parallelepiped.
(No, none of the terms are "trirectangular tetrahedron." Then again, since a trirectangular tetrahedron is a pyramid, it would fit better with tomorrow's lesson.)
Let's get back to Euclid's Elements, since I've started discussing it yesterday. We can finally look at the first two definitions in Book XI:
https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html
Both Euclid and the U of Chicago distinguish between the boundary and the interior.
Here are the definitions of the two surfaces mentioned in the lesson title:
It's interesting to compare Euclid's definitions to the U of Chicago's. Euclid imagines a cylinder as a solid of revolution -- we take a rectangle and rotate it around one of its sides. (Don't forget that a rotation in 3D has an axis, not a center.) But in the U of Chicago text, we perform a very different isometry -- we begin with a circle and translate it out of the plane. David Joyce points out that Euclid's definition doesn't generalize -- it describes only right cylinders. Not only does the U of Chicago definition include oblique cylinders, but the idea of cylindric solids (or surfaces) extends to prisms as well.
By the way, Euclid defines a prism in terms of its faces. Since the lateral faces of Euclid's prisms are parallelograms (and not necessarily rectangles), oblique prisms are included.
Let's return to the theorems. Since we looked at Proposition 4 yesterday, let's try Proposition 5 today:
Here's a modern version of the proof. Notice that Euclid writes "For suppose they do not...," which implies that this is an indirect proof.
Given: Line AB perpendicular to BC, BD, and BE.
Prove: Lines BC, BD, and BE are coplanar.
Indirect Proof:
Assume to the contrary that they aren't. By the Point-Line-Plane Postulate part f, any three points are coplanar, and so let P be the plane containing B, D, E, and let Q be the plane containing A, B, C.
By Point-Line-Plane part g, since planes P and Q intersect (in B), they intersect in a line. Let F be another point on this line. So lines AB, BC, and BF are coplanar as they all lie in plane Q.
We are given that AB is perpendicular to both BD and BE, and so by yesterday's Proposition 4, AB is perpendicular to the entire plane containing BD and BE, namely P. By the definition of a line perpendicular to a plane, AB is perpendicular to every line in P through B, which includes BF.
But we are given that AB is also perpendicular to BC. Therefore AB is perpendicular to two lines, BC and BF, both in plane Q. This is a contradiction, since through a point on a line, there is exactly one line in the plane perpendicular to the line. Thus the assumption that BC, BD, and BE aren't coplanar is false. Therefore BC, BD, and BE are coplanar. QED
This proof isn't as difficult as yesterday's, but it is an indirect proof -- and we don't cover indirect proofs in our text until Lesson 13-4. It also requires yesterday's Proposition 4 in order to prove -- and Proposition 4 has a difficult proof.
Here is the worksheet for today's Lesson 9-2:
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