I need a 2-dimensional pattern for using 3 colors

(I also need a better title for this post.)

After two years of knitting, which included about 8 months of not knitting and then 3 months of knitting every spare second in order to finish, I’ve almost completed 49 squares for an afghan.  Now I just have to put it all together.

So here’s what I have:
24 squares in single colors (8 each of Blue, Green, and Yellow)
25 squares in multi colors (roughly 8 of each pair of colors, although some use all three colors)

I want to put them into a 7×7 grid in a way that alternates single and multi-colored squares, and while I can do this according to trial and error I feel like there should be a better way.  The end result would look something like thing:

(This is the last time I made an afghan, which also took me 2 years to do.  Apparently I like this triple of colors, though I’m using a darker blue and a lighter green this time.)

There ought to be a pattern of how to lay out the squares, probably alternating single and multi-colors, so that the colors are more or less evenly spread over the entire blanket.  There ought to be LOTS of patterns, I think — and probably patterns that generalize to using n×n squares of k colors [or even n×m squares].   Does anyone see anything, obvious or not?  Ideas for where to look would be most welcome!

I’m aware of the irony that my reason for wanting a patterns is to save time for the initial setup, even if I end up switching some stuff around, but I’m spending even more time than I’d save trying to look for a pattern.  Still, in 2+ years if I face this same question again I’m sure all that work will pay off!

The Godzillas make Chocolate Éclair Pie

No math today (except the usual fractions in cooking):  the purpose of this post is to introduce you to Godzilla’s young ward, who currently (but not necessarily permanently) goes by the name Mini-G.   Little Godzilla is not very tall:

but this means he gets to travel more, and do things like ride the bumper cars at amusement parks.

Today (well, actually, last weekend) Big-G and Mini-G teamed up to prepare a delectable feast of a dessert that is dangerously easy to make:  Chocolate Éclaire Pie, the recipe of which is due to our no-longer-brand-new faculty member Nicole.  (Actually, I think the recipe comes from her mom, and possibly someone else before that.  Thanks Nicole’s mom and anyone else it might have come from!)

Here’s how you start:

1 box of Graham crackers
2 small boxes of vanilla instant pudding
1 (8oz) carton of Cool Whip
3.5 cups of milk

Grease the bottom of a 9×13 cake pan (so maybe this is really a cake and not a pie) and line it with graham crackers.  You’ll use just under 1/3 of a box, so you’ll have some left over for snacking.

Mix the pudding with milk and beat it about about 2 min.

Then blend in the cool whip. You’re practically done with the cake-part now!

(There’s about an 80% chance that someone was singing, “Whisk it!  Whisk it good!” during the filming of this portion.)

Pour 1/2 the mixture over the crackers. Add another layer of crackers and cover the rest with pudding mixture, then add a final layer of crackers to the top of that.

Cover it, and refrigerate it at least two hours.  When it’s about ready, it’s time to make the Topping!

¾ cup cocoa
¼ cup oil
4 tsp vanilla
3 cups 10x (powdered) sugar
6 Tbsp (3/8 cup) milk
6 Tbsp butter or margarine, melted
4 tsp white karo syrup

Mix all the topping ingredients together until smooth.  [An electric mixer would have been more effective, but harder for the Gs to hold.]

Spread on top of the graham crackers, and refrigerate it for 20 more hours.  And now it’s ready!

YUMMMMMMMMM!!!!

Note 1: The original recipe only had half the amounts for the topping, but Nicole confessed that she always doubled it, so who am I to argue?

Note 2: When I made it the topping came out a little thick, so it might be OK to add a little more milk if it seems that way to you too.

Note 3: I’m not totally sure about that final 20 hours; I keep meaning to double check that it’s not a typo.  But I kind of like the 20 hours because it means you have to make it a day in advance, which works well if you’re having people over or bringing it somewhere.

A Newsworthy Ha’penny

Here’s a good rule of thumb:  if you’re trying to calculate how much money to send an insurance company, it’s probably a good idea to round up.  That’s a lesson that La Rosa Carrington learned the hard way.

Carrington had health insurance under her job, and when she lost her job she was allowed to continue her health insurance under federal COBRA law.  Trouble is, she didn’t get a bill so she estimated the amount she would have to pay:  her payments were “a little over $471.87  per month” (according to The Gazette in Colorado Springs, where the story first appeared on July 6)  but because of the 2009 American Recovery and Reinvestment Act she only had to pay 35% of that.

Carrington didn’t get a bill from Discovery Benefits, and yet she knew it was important to keep up the payments, especially because she was also undergoing chemotherapy for leukemia so details like current health insurance coverage  were totally non trivial matters.  She sent them a check for  $165.15.  Trouble is, Discovery Benefits said she owed $165.16,  and canceled her coverage.  She called, they refused to budge, and finally the supervisor did the calculation herself and decided that rounding the amount to $165.15 was actually right, or at least reasonable, and the penny was paid [either by the company or by a person in the company; it's not clear which].

So be warned:  sending in that extra penny might be good insurance for your insurance.

The story could end there, since rounding is all mathematical in and of itself, but there’s a tangent that I’m still wondering about:  what’s the deal with the monthly payments being “a little over” $471.87 each month?  If the annual dues were $5662.46, for example, then the monthly payments would be $471.8716666…, which would round to $471.87, but 35% of  the original $471.871666….. would actually be $165.155083333… which does round to $165.16 using conventional models of rounding.  It seems plausible to me that the Benefits Computer was just rounding, and not necessarily rounding up all the time, and that the multiple rounds gave a difference of a penny, which would make this a story not about rounding up versus rounding down, but about the compounding of rounding errors.  I looked at a few different reports on this, though, and never saw mention of this so it’s possible that the Benefits Computer was automatically rounding up for all rounding as implied.

Just how *is* luggage volume calculated, anyway?

Last week I posted about a mistake in the translation from Imperial to Metric units on a luggage website, and Cathy Campbell pointed out in the comments another possible error:  that the length times width times depth doesn’t equal the capacity.  In this example:

25×18×11 equals 4950, not the 4720 cubic inches of capacity advertised.  Possibly this is due to the thickness of the materials, but I’m wondering how they take that into account if that’s what it is.  It seems more likely to me that they’re using a formula to get capacity than measuring it physically, but I’m not sure what formula they’re using.  Anyone want to try it out?  (It might be trivial or complicated; I spent more time uploading screenshots than actually working on a formula, although I did notice some oddities in the process.)

Here’s one line of luggage in this brand (the same as above), with the metric units removed because, well, they’re not all correct:

Here’s another line (same brand):

And a third line in the same brand.

Anyone want to hazard a guess as to the (possibly existing) formula(s) (now, with added parentheses!)?

Wanted for Insurance jobs: A few good mathematicians

Probably more than a few, really.

There have been a few Math Mistakes in the News this summer with regard to insurance companies, though unfortunately I can’t seem to dig up what all of the actual mistakes are.

Back on May 5 the California Department of Insurance released a press statement that Anthem Blue Cross had been making some math mistakes and they were going to be under extra scrutiny.  According to the statement:

The errors identified included:

• Error #1: Double counting of aging in the calculation of underlying medical trend for the projection of total lifetime loss ratio.
• Error #2: Anthem overstated the initial medical trend used to project claims for September 2009 for known risk factors.

Both of these errors are errors of math and not differences in actuarial opinion.

I didn’t see anything about this costing a particular amount of money, though a June 25 article from the Los Angeles Times indicates that they canceled a rate increase of up to 39% for many of their California customers customers as a result.

Then, less than two months later, Aetna Inc. also had some math woes.  According to the same LA Times article,

Connecticut-based Aetna Inc. had sought an average 19% increase in rates for its 65,000 individual customers, but pulled back after multiple math errors in its paperwork were found by its own staff and by an independent consultant working for the state.

I was tempted to write “Bummer” but there’s really nothing bummerish about not having a 19% rate increase.  There’s no direct statement of what the mistakes are, just that “There were multiple errors … in the way [Aetna] annualized premiums and in the compounding of the rate increase,” according to California Insurance Department spokesman Darrel Ng.

Another article that same day on a Consumer Watchdog site, quoted Watchdog president Jamie Court as saying, “It’s amazing how insurers are making mathematical errors when they’re not used to regulators checking their math.”

Incidentally, a similar error was more recently discovered across the pond.  According to citywire,

Yorkshire and Clydesdale Bank today said it is in the process of mailing around 18,000 variable rate mortgage customers to apologise for miscalculating their monthly repayments and to suggest ways customers can repay what they owe.

The bank said the calculating error, which was exacerbated by last year’s unprecedented falls in interest rates, led to the bank collecting less than the contractual minimum monthly payment required for customers to pay their mortgage within their agreed term.

Moral?  Check your math.

Seriously, though this topic is serious enough, several of our math grads in recent years have gone into different aspects of insurance and finding errors can be a pretty important part of their jobs, whether it’s part of the official description or not.

How common are pentagon buildings/rooms?

I read a news article that a 4th century Roman villa was recently [where recently might mean 4+ years ago] discovered near Aberystwyth in Wales.  According to this July 26, 2010 article from the BBC news, “It was roofed with local slates, which were cut for a pentagonal roof.”

Pentagonal roof?  That sounded really cool, though I wasn’t quite sure what it meant and assumed it referred to the shape of the building itself.  The article included an outline of the area, but I couldn’t really tell if it was a pentagon or not.

In searching some more, however, it turns out that the pentagonal refers to the individual pieces of slate.  In the photo on this Heritage of Wales site they explain:

Two pentagonal Roman roof slates from the Abermagwr villa, the one on the right nearly complete. Made from local stone, with square nail holes, these slates constitute what may be the earliest slated roof in mid Wales.

Dang — no pentagonal rooms after all [as a footprint on the Heritage site confirmed], though I bet the roof looked cool anyway.

(Though this roof is from France, published by Arlette1 under GNU-FDL.)

Still, the pentagonal roof question got me wondering how common 5-sided buildings are.  There’s THE Pentagon, of course:

But there are some older examples.  There’s a stone age temple in Sweden according to this article from the Archaeological Institute of America (which actually claims that it is “a near perfect pentagon”).

I also ran across a pentagon room from Flatland by Edwin A. Abbott:

but I feel a little guilty posting it because I haven’t managed to finish the book.  And in it’s current exhibit our Science Museum has a photo of a building that isn’t the Pentagon, but there aren’t any captions so I don’t know what it is.

I had a little more luck with pentagonal rooms.  The  1903 book Roof Framing Made Easy:  A Practical and Easily Comprehended Construction, Adapted to Modern Construction, for Laying Out and Framing Roofs by Owen B. Maginnis has a whole chapter on pentagonal roofs, beginning on page 82, although Owen confesses that “this roof is of a form rarely met with in building construction”.  Incidentally, despite Owen’s claim to make it as simple as possible, I’m thinking simple in 1903 is a tad different than simple in 2010.  [Incidentally, Owen also has a chapter on hexagonal roofs and no fewer than FOUR chapters on octagonal ones, but the heptagonal roof is skipped completely.  Poor heptagon.]

Still, as Owen pointed out, they aren’t exactly common and it makes me wonder if there really just aren’t all that many, or if I just haven’t looked.

Luggage shopping: Find the error

We’ve been luggage shopping recently in honor of an upcoming sabbatical (and in recognition of the fact that pretty much every piece of luggage we own is ripped:  turns out that the combination of buying the cheapest possible suitcases and cramming them as full as possible is maybe not the best for their long-term health.)  I was looking up different kinds of suitcases, and after staring at too many numbers for too long, I suddenly noticed something weird.  The error as near as I can tell is systemmatic — it was in in the stats for every piece I looked at for this company.  Here’s an example (chosen deliberately from the newest model line, since I initially wondered if it had been fixed).

Can you find it, and tell what the (web designer?  programmer?) did wrong mathematically?

[I actually found this a couple weeks ago and send them an email to let them know, but haven't received any reply and as of today nothing had been changed online.  I suspect it's buried in someone's mailbox somewhere.]

The photo in the upper left is actually of a piece of artwork called Travellers Garden by John Grimshaw at the Old Mangotsfield Station.  The photo, licensed by CC Atribuition-ShareAlike 2.0, is by Linda Bailey and can be found, of course, on Wikipedia.