The on-going struggle, 1


The on-going struggle to get my Asus Zenpad 8 to do what I want. Part 1.

Sigh, the more I use this thing designed around Android, the more I want to kick myself. What should be simple tasks aren’t supported, and the tasks I want that ARE supported either aren’t in the Asus user manual, or don’t work the way the manual indicates.

1) Adding HTML files
I like working with shortcut pages that I write myself that are HTML-based. Since I teach English to adults, I need a spelling dictionary. I don’t like the apps in the Google Play store, and I often don’t have wifi connections in the class room, so I put together something myself where one page has links to a bunch of other pages, all on internal storage. I added a 64 gig memory card, and naturally that’s where I wanted to put my spelling dictionary (the normal memory card is where all the apps get stored, and that’s only 10 gig). Surprise, surprise, the links didn’t work. They worked fine on my laptop when I beta-tested everything, but I kept getting page not found errors when I tapped on the link in the HTML index page on the tablet. The individual pages would run if I tapped on them directly from file manager, but I couldn’t link to them from one HTML page to the next. Then I moved all the files to a different directory on the internal memory card, and suddenly everything works fine.

This isn’t documented anywhere, and there’s almost no discussion of the problem on the net that I can find. It’s nice that the links work now, but the point is to NOT clutter the 10 gig card with something that could easily be stored on the 64 gig card. It also means that any other HMTL pages I write will have to go on the smaller card as well, which makes no real sense.

2) Customizing the Home Screen
I get that computer suppliers want to package their products with their own software recommendations, and that novice users might find these apps useful. But, I don’t want them. I want the apps that I want to use placed on the home screen in the way that I can get to them most easily. Such as with the index page for my above HTML spelling dictionary app. So, I finally get my HTML app working and now I want to add a shortcut to it. I go to the Asus documentation, and it’s telling me to select Home Edit, and then Edit Page. Guess what? The Home Edit screen doesn’t have anything marked “Edit Page”.

I go to the net and start searching around, and I find a mention of touching an icon and holding it for a couple of seconds until the screen changes. The new screen has a trash icon, so I can drag any shortcuts I don’t like to the trash and get rid of them that way. I also discover that I can press and hold an icon, and when the screen changes I can drag the icon left or right to put it on a different screen. Swiping the screen right twice takes me to a blank screen with a “+” sign on it. Tapping on that gives me a new blank screen, but I’m good with just two screens right now. What I want is fewer icons, icons just to the apps I’ve downloaded from the Play store, and shortcuts to a couple movies and my new HTML pages.

So, naturally, Android doesn’t support adding new shortcut icons. Yes, using the above tap and hold action on an existing icon I can delete something I DON’T want, but I can’t add new shortcuts without downloading yet another app from the Play store. I’ve been finding Tom’s Guide to be helpful for questions like this, and they suggest using ES File Explorer File Manager. I downloaded that (which of course takes up more space on the 10 gig card), and it does let me add shortcuts to the desktop for the files I want. But the icons for the new shortcuts are all the same, and it’s ugly. I’d like to be able to specify the icon to use, like a musical note for a music file, or a spider for an HTML file. Still, it’s better than nothing.

3) Updates
My Asus tablet apparently was installed with software from 2016, and now that I’m using it with wifi from a coffee shop, it tells me that there are over 40 updates pending. The first time I tried downloading the updates, the Zenpad got about 8 files in and then complained that it couldn’t install the rest. A couple days later, when I was back in the coffee shop, I got another notification of 40+ updates. This time, 30+ downloaded and installed, but the Zenpad choked on the Gmail accessory, and none of the subsequent updates installed because they’re useless crap tied to messenger and contact functions that are tied to gmail. I have logged into gmail with the Zenpad, but that apparently wasn’t enough. I don’t want the remaining updates but, it is just another annoyance.

4) Screensaver and power loss
When I was struggling to get the above HTML files running right on the tablet, I had it connected to my laptop through the USB port, and the tablet kept sleeping after one minute. I swept up on the home screen, went to System settings, and eventually figured out that I need to tap on Display, then Sleep. This let me change the sleep timer to 5 minutes, and I figured that that was that. But, the next time I used the tablet after putting it away for the night, the battery was half discharged. My mistake was in thinking that just closing the cover of the tablet holder and leaving it alone would mean that the thing would go into sleep mode after 5 minutes. But, no, the holder cover was touching the screen, mimicking user activity and preventing it from sleeping. Now, when I put it away, I have to press the power switch on the side quickly to force sleep mode. (The reason I don’t want the screen to go black when I have it connected to the laptop is that I’m using a lock password, and if it does sleep, the only way to wake it back up is to press the power switch (touching the screen has no effect) and reentering the password all the time.)

5) Wallpapers
This may seem obvious, but there are two actual wallpaper modes. The first is when you enter the password when you turn the tablet on, and exit from sleep mode. The second is when you actually use the tablet normally. If you pick one or the other mode when you select a wallpaper image, you get the wallpaper for that one mode but not the other. Yes, it’s kind of obvious, but not intuitive. If you choose “Home and lock”, you get just the one wallpaper for both cases. Which to me makes the tablet less confusing if you know what to expect in advance when choosing wallpaper files.

However. Wallpaper files have to be in the Gallery or Wallpapers directories, which are both on the 10 gig card, and that takes up some of the limited space there. There doesn’t seem to be an inherent way to browse the external SD card. The solution seems to be to only have one or two images in the Gallery directory at a time, and copy any new files you want to switch to from the external card when you want them, and delete the old files.

Also, if the wallpaper file is large (mine are 1300×760 for my laptop layout), they’ll have to be cropped. I’m not sure yet what the actual screen dimensions are, but they seem closer to 640×480. And, the default font colors for the time and weather information are white on transparent. If the wallpaper is too “noisy” or all white, the information display on the screen, and/or the other icons are going to be difficult to impossible to read. Swiping up and tapping Home Edit lets you change the Icon label color, but that’s just for the app names that appear under the icons on the screen. The text color for the Time, Weather and Location information will still be the default white. Tapping on the Time or Weather information does bring up the settings screen for them, but I haven’t found any way from that screen to change the text color.

In short, simple wallpapers that aren’t all white seem to work best. I have a red and black checkerboard pattern that came from the skin-horse.com webcomic that I’ve settled on.

6) Sudden loss of screen brightness
This one took me a while to figure out. I’d be using the tablet and suddenly the screen display would dim. Then it would become bright again. Then dim again. I was thinking the thing was broken, and I was debating whether I’d have to bring it back to Bic Camera for replacement. Then it dawned on me (yes, there’s a pun there) that the problem happened whenever I had my hand or head over the screen. Specifically, over the primary camera. Apparently, the Zenpad uses the camera to automatically adjust the screen brightness based on the lighting in the room. And, because I use my right hand to hold the tablet steady when I’m working, I’m just naturally putting it directly above the camera, and that makes it look to the Zenpad that it’s in a space with dim lighting. The fix is to up sweep the home screen, tap System settings->Display->Brightness level. And then uncheck the Automatic Brightness box. It does mean that I have to manually change the brightness level now, but generally I prefer having it set at “Bright” all the time.

Wednesday Answer


Solution: What is the smallest convex area in which a line segment of length 1 can be rotated 360 degrees? (A convex figure is one in which a straight line, joining any two of its points, lies entirely on the figure. Examples include circles and squares.)

Answer, an equilateral triangle of height 1. For the line segment to rotate, the sides have to be at least length 1. Of all convex figures with widths of 1, the equilateral triangle has the smallest area. Try taking a toothpick and a cardboard cutout of a triangle and check for yourself. To make it easier, glue a second toothpick in the middle of the first one to make an axle for rotating the first toothpick within the triangle cutout.

Colossal Gardner, ch. 4


We now move into Plane Geometry and the concept of Curves of Constant Width. Actually, the main content of Gardner’s article is captured in the wiki entry. His starting point is that you can use Reuleaux triangles in the place of wheels in a roller-based system, and a platform placed on top of the rollers will remain level and steady as the rollers rotate across a flat surface. So, it’s not necessary to use circular wheels if you don’t want to. (If you want pictures, go to the wiki article.)

The more interesting application of the Reuleaux triangle was in the Watts Brothers Tool Works patent for a drill and chuck system capable of drilling square holes. They also manufacture drill bits for pentagonal and hexagonal holes that have sharper corners. Although the outer edge of a curve of constant width is in contact with its bounding space at all times (as with a square for the Reuleaux triangle), the center of rotation moves around all over the place. That means that for the Watts square-hole bit, a special chuck is required to trace out the correct rotation to ensure that the hole actually comes out square. You can see the patents (filed in 1917) at Google patents: US1241175, US1241176 and US1241177.

The puzzle this time is based on the Kakeya Needle problem. What is the smallest convex area in which a line segment of length 1 can be rotated 360 degrees? (A convex figure is one in which a straight line, joining any two of its points, lies entirely on the figure. Examples include circles and squares.) Check the wiki article on Kakeya sets for an illustration of the Kakeya Needle.

 

Subways of the old Romans


This is very cool. Someone unearthed one of the subway maps used by the Romans. Now we know why they were able to run their Empire so efficiently. And a one-month rail pass cost only IV denarii.

Wednesday answer


Wednesday answer:
Find a copy of J. A. Lindon’s “Doppelganger“.

 

Colossal Gardner, ch. 3


The last chapter of the arithmetic and algebra section is on palindromes. I absolutely love the “near miss” palindrome at the front of the article, attributed to Ethel Merperson, in Son of the Giant Sea Tortoise, edited by Mary Ann Madden (Viking, 1975) –

A man, a plan, a canal – Suez!

Palindromes can take many forms, from words and sentences that can be read the same forward and backward, numbers that can be rotated, palindromic primes, and even photos of things (like a bird in flight, going from wing tip to wing tip).

Here’s a game. Start with any positive integer. Reverse it and add the two numbers together. There’s a conjecture that you’ll get a palindrome after a finite number of steps.

47
74

121  <— End

183
381

564
465

1029
9201
—-
10230
3201
—-
13431 <- End

There have been papers written on the existence of palindromic primes and powers. You can play with palindromic roots to get palindromic squares, such as 121^2 = 14641.

You can find many of the same language examples in the wiki article. Yreka City in California used to have the Yreka Bakery and Yrella Gallery. Then there was the former premier of Cambodia, Lon Nol. And, you can have sentences where the word order is palindromic: from J. A. Lindon – “You can cage a swallow, can’t you, but you can’t swallow a cage, can you?”

Note: Lindon was a pioneer in the recreational mathematics field of anti-magic squares. In magic squares, the rows and columns all add up to the same number. With anti-magic squares, the sums of the rows, columns and diagonals are all different. He died in 1979.

Challenge: Can you find a copy of J. A. Lindon’s “Doppelganger,” and can you write a longer palindromic poem yourself?

Comments: I love palindromes, and I had a book collection of them at one time. Back when I first had a 4-banger calculator, I’d tried to find palindromic numbers. I’ve never heard of anti-magic squares before, but they could be fun to play with.

Hachette 3D Puzzle Series, vol. 9


(All rights belong to their owners. Images used here for review purposes only.)

Well, it’s been a couple months since I first wrote about Hachette’s new serialized 3D wood puzzle magazine series. Of the 9 puzzles so far, I only had an interest in #1 (because it was the cheapest one, and I wanted to see what the magazine looked like), and #6. Unfortunately, for some reason #6 is the one that sold out right away and I haven’t seen a copy in the stores at all. These things do run $15 each, so it’s not like I really feel a desperate need to buy it over the net, but if I held the box in my hands in the store, I’d probably get it. (#6 is a 3x3x3 wooden cube slightly rotated and designed to set within a restraining dowel peg holder.)

I also liked the look of #9, so I bought the only copy still at the Junkudo bookstore here


(Ad for the display shelves.)

At first glance, “Mysterious Ball” is a puzzle in that there’s nothing saying what you’re supposed to do with it. I pushed and pulled the pieces sticking out of the ball with no effect. Then I checked out the magazine to see what was in it. One of the first pages gave a hint, and another showed the thing fully disassembled. I didn’t even bother looking all that closely at the photos. 30 seconds later, I had it apart, and back together again. I spent another 5 minutes really examining the MB and making sure I knew exactly why the parts moved sometimes and not others. And now, yeah, I can confidently claim that I’ve got it mastered. It’s got a difficulty level of 3 out of 5 stars, which is probably overstating things a bit. I’d say that it’s an “Easy.”


(Philidor article.)

The magazine feels kind of short this time. While the full thing is 20 pages, the outer sheet is detachable and is primarily advertising for the full series, with a page dedicated to the clear shelving structure you can buy for 5,000 yen ($48 USD) to display all the puzzles in your collection. That leaves 16 pages for the actual magazine itself, where you get the front cover, the inside front cover with publication information and table of contents, 2 pages describing the Mysterious Ball, 5 pages of text puzzles plus their solutions, 2 pages on Francois-Andre Danican Philidor, 1 page for the classic painting memorization game, and 4 pages detailing the solution for the Gyro puzzle from volume 8 (which I used to have a long time ago, and felt no need to buy again). If you’re keeping track, that’s just 7-8 pages of useful material (4 pages of text puzzles, the memorization painting and the piece on Philidor). (The solution for the Mysterious Ball will appear in vol. 10.)


(Look at this painting for 1 minute. Then look away, and answer the questions about it.)

Philidor (1726-1795) was a French composer, best known musically for his comic opera, Tom Jones (1765). However, he may have found a longer-lasting fame as a chess player. He’s considered the best player of his age, and had been friends with Benjamin Franklin. He wrote one of the most definitive books on chess, “Analyse du jeu des Echecs” (1749), with other editions in 1777 and 1790, and it went through another 70 editions and had been translated into English, Russian, German and Italian by 1871. He was the first player to recognize the importance of the pawn in the game. The Philidor Defense, Philidor’s Legacy and Philidor’s position are all named after him. In 1882, Amedee Dutacq (music) and Abraham Dreyfus (libretto) premiered a one act opera-comique entitled “Battez Philidor,” in which an impoverished musician has to beat Philidor at chess in order to win the hand of his sweetheart. Although Philidor agrees to throw the match, he gets distracted and wins by accident.


(Example page for the text puzzles.)

Summary: I like the Mysterious Ball puzzle and plan to challenge my students to solve it. I found the article on Philidor to be interesting, and I plan on tackling the remaining text puzzles at some point.

The next puzzle is going to be the cube snake, scheduled for a June 21 publication. This is the same elastic band wrap-around puzzle that I got from the capsule ball dispensers for $2, so I’m not going to buy volume 10. #11 is the Tower of Hanoi. #12 is the “Devil’s Star” (also from the capsule ball series). #13 is “Turn Table,” another scramble/descramble game in the vein of Rubik’s Cube. It’s pretty, so I might get it, although I’m useless with the Rubik’s Cube, so maybe I won’t. #14 is the White Wood Pig, one of the rare 3D burr puzzles that I kind of want to get. #15 is the Devil’s Box, one of only 2 puzzles rated 5 stars. It looks nice, so I might get that. #16 is a set of wooden disks with color markers that you’re supposed to align so the colored dots match up on all disks. It’s kind of a Penrose tile game, so I might get that, too. #17 is the Cross Piece Cube, which is similar to one of the other capsule ball puzzles. And #18 (the series might go past 18 volumes, but there’s no confirmation of that) is the Devil’s Molecule, which also looks pretty. At the moment, I’m mildly interested in #13 and #16, slightly more interested in #14 and #17, and will probably definitely get #15 and #18. But, these are coming out one every 2 weeks, and, again, they’re about $15 apiece. Even if I do get the ones listed here, the first one won’t hit the shelves until August. See you then.

Ch. 2 Solutions


Number of regions that can be made for a pancake of n straight cuts:
1/2 * n * (n-1)

Number of designs for n beads of 2 colors:
There’s no way to solve this with finite differences. It has to be approached with a recursive program.

Number of triangles for n straight lines:
You need 3 lines to get one triangle, so
0 1 2 3 4  5 : Lines
0 0 0 1 4 10
0 0 1 3 6
0 1 2 3
1 1 1

Assume that no two lines are parallel, and no more than two intersect at the same point. No three lines can form more than 1 triangle, and any set of three lines must form one triangle. The problem becomes similar to: How many different ways can n lines be taken three at a time. The answer can be found using Newton’s formula, as 1/6*n*(n-1)*(n-2).

Comments: Finite differences can be fun. Pick something you can measure at fixed intervals (car speeds or distances, average numbers of cars on the freeway during the day, prices of haircuts at different hairstylists) and see if you can find patterns for them as a factor of n.

Colossal Gardner, ch. 2


Chapter 2 is a continuation of the section on Arithmetic and Algebra.

This time, we get the Calculus of Finite Differences. This concept was first published between 1715 and 1717 by Brook Taylor, who developed the Taylor Theorem. Finite differences are an early form of integral calculus, and the idea is to look at the results of an equation in steps, effectively taking the differences of that equation at those steps to determine the underlying properties of an event. It’s not an infallible way of building up equations to explain specific behavior, but it’s a good starting point in many cases.

Gardner describes a “mind reading” party trick used by W. W. Sawyer, who taught math at Wesleyan University. You ask someone to think of a quadratic equation, with no powers greater than x^2 (to keep things simple).

For example, use 5*x^2 + 3*x – 7.

You ask the person to give you the answers for x = 0, 1 and 2. You can turn your back on them so you can’t see them working the math. In this case, the answers would be -7, 1 and 19. With practice, you could do the following steps in your head in a couple seconds.

The process is to write out the answers in a row:
-7 1 19

Subtract the number on the left from its neighbor to the right and put the results on the next row:

(1 – -7) = 8, and (19 – 1) = 18
8 18

And do the same thing to build up the third row.

18 – 8 = 10

-7 1 19
8 18
10

The original equation can be built up by the following rules:
The coefficient for x^2 is 1/2 of the bottom number.
The coefficient for x is the first number of the middle row minus half the bottom number.
The constant is the first number of the top row.

a*x^2 + b*x + c
a = 10/2 = 5
b = 8 – 10/2 = 3
c = -7

The calculus of finite differences is a study of these kinds of equations. It had been further developed by Leonhard Euler and George Boole (creator of boolean logic) but fell out of favor in the 1800’s. In the later 1900’s, it became useful again for use in statistics and the social sciences. You can determine the gravitational constant this way. Time the fall of a stone and record the distances at one second intervals:

0 16 64 144 256
16 48 80 112
32 32 32

distance = 32/2 * x^2 + (16 – 32/2) + 0 = 16*s^2

There’s no guarantee that this equation holds absolutely in all cases, but it’s one way to start approaching the development of a theory.

I’d done something like this myself many years ago when I was bored one day, looking at the differences of different strings of powers:

1 4 9 16 25 36
3 5 7  9  11
2 2  2  2

1 8 27 64 125 216
7 19 37 61 91
12 18 24 30
6  6  6

So, I’d discovered finite differences without having any idea what it was. Note that as you get higher powers of x, you get more lines on the pyramid.

Gardner gives a couple additional puzzles to play with – What’s the maximum number of pieces a pancake can be cut into by n straight cuts? How many different designs can you get when making circular necklaces of n beads of two colors (the beads are white or black). And, what’s the maximum number of triangles that can by made with n straight lines?

 

Coconuts answer


There are actually two variants for the coconuts puzzle. In the older, easier one the monkey gets an extra coconut at the end. Versions of this one apparently go back to the middle ages. The newer version used by William, where the monkey only gets 5 coconuts, is more difficult to solve.

The Diophantine approach is to build up a polynomial as follows (for the older version of the puzzle):
N = 5A + 1
4A = 5B + 1
4B = 5C + 1
4C = 5D + 1
4D = 5E + 1
4E = 5F + 1

N is the original number of coconuts, and F is the number that each of the 5 sailors received at the end. The way to read this is, at the beginning, sailor A divided the pile “N” 5 ways, kept one pile of size “A”, and had 1 coconut left over that he gave to the monkey. Now, there is a pile of 4*A coconuts that is divided 5 ways by sailor B. He keeps one pile of size “B”, has one coconut left over for the monkey, and puts the other four B-sized piles back together again, etc.

If we turn this into one equation by normalizing A, B, C, D and E (i.e. – E = (5F+1)/4; D = (5E+1)/4 = (5F+1)*5/16; etc.) we get 1,024*N = 15,625*F + 11529.

We want the smallest positive value of N such that both N and F are integers. This is easy to do in VBScript as a simple program to run N from 1 to 10,000, but not so easy to derive by hand. (My solution is N = 15,621, F = 1,023.)

One of the interesting things about the Diophantine equations is that some of them have only one solution, others have infinite solutions, and others have none. In our case, the coconut problem has an infinite number of solutions, so we want the smallest one.

However, Gardner presents a twist approach using negative coconuts, which dates back to Norman Manning in 1912. Alternatively, you could take some of the coconuts and paint them blue to be easier to track. The reasoning goes – if you’re dividing up the coconuts into 5 equal piles 6 times, with no leftover coconut for the monkey at the end, then the smallest number that would work is 5^6 = 15625. Now, take four of those 15,625 coconuts, paint them blue and put them in the bushes. This leaves you with 15,621 coconuts, which you CAN divide into 5 piles, with one for the monkey for round 1. Put the four blue coconuts plus the other 4 piles together to make one pile of 5^5 coconuts. This can obviously be divided by 5. But, we pull the 4 blue ones and give the extra to the monkey. In Gardner’s words “This procedure – borrowing the blue coconuts only long enough to see that an even division into fifths can be made, then putting them aside again, is repeated at each division.” After the last round, the blue coconuts are left in the bushes, not belonging to anyone.

The use of blue, or negative, coconuts explains something I discovered when I ran my program. One of the solutions is for N = -4, and F = -1. Adding 5^6 (15,625) to this N gives 15,621, which is the next solution my program found, with f = 1,023. All subsequent solutions have the form N = k*5^6 – 4, and F = k*4^5 – 1 (where k runs from 1 to infinity).

The above description is for the variant of the puzzle where the monkey gets that 6th coconut in the final round of dividing the coconuts into 5 piles. For William’s version, where the last round doesn’t have the extra coconut, the Diophantine equation is:
1,024*N = 15,625*F + 8404

There’s a different general equation that can be used here, depending on whether “n”, the number of sailors, is even or odd:
# Coconuts = (1 + nk)*n^n – (n-1)  : for even n
# Coconuts = (n – 1 + nk)*n^n – (n-1) : for odd n

k is the multiplier used above to get infinite solutions as multiples of N, and for the lowest positive solution, k=0.
For n = 5 sailors,
N = # Coconuts = (1 + 0*5)*5^5 – (5-1)
N = 1*3125 – 4
N = 3,121
The last round of divvying will give 5 piles of 204 coconuts, and nothing extra for the monkey.

Finally, the last puzzle on Monday had three sailors finding the pile of coconuts. The first sailor takes 1/2 of the pile, plus half a coconut. The second sailor takes half of the remaining pile plus half a coconut. The third sailor does the same thing. Left over is exactly one coconut, which they give to the monkey. How many coconuts did they start with?
If you work backwards,
(1 + 1/2) * 2 = 3
(3 + 1/2) * 2 = 7
(7 + 1/2) * 2 = 15
Answer: 15 coconuts

Comments: Man, I never expected this chapter of the book to give me so many problems. I’d initially thought I’d spend an hour writing it up and that’d be the end of it. But, I kept making mistakes in forming the Diophantine equations for each puzzle, and both my math, and my VBScript program kept coming out wrong. I spent close to 2 days on this one.

Obviously, I’m not as good at recreational math puzzles as I’d liked to think I was.