Archive for January, 2019
The Other Golden Ratio
You’re a math geek. I know that to be true, because you’re reading this blog. And I also know that when you hear golden ratio, you think of this:
Or this:
But there’s another, different — and, dare I say, better? — golden ratio that may be even more important to learn. Especially if you’re one of the mathy folks to whom this adage applies:
Wherever you find four mathematicians, you’ll likely find a fifth.
As Homer Simpson says, “It’s funny ’cause it’s true.”
Like many classroom teachers, I’m often ready for a cocktail on Friday afternoons. And like those teachers, I don’t want to spend a lot of time thinking about it; I don’t want to rummage for a recipe; I just want to relax and have a drink. But, I also don’t want to have the same cocktail every Friday; variety is the spice of life.
So, what’s a boy to do?
Simple. Follow the advice from the folks in the Food Hacks division at Wonder How To, who claim that the following ratio will yield a delicious cocktail every time:
2 : 1 : 1 :: alcohol : sour : sweet
Right now, you’re probably scratching your head and thinking, “Can it really be that simple?”
I’m here to tell you, friends — it is.
For instance, 2 parts tequila, 1 part lime juice, and 1 part triple sec? That’ll get you a tasty margarita.
And 2 parts bourbon, 1 part lemon juice, 1 part simple syrup? None other than a classic whiskey sour.
If you combine 2 parts tequila, 1 part rhubarb liqueur, and 1 part malic acid — it’s what gives green apples their tartness — then you’d be getting close to a drink they call The Scarlet Lantern at Bar Congress in Austin, TX. (They also mix in black cardamom-strawberry shrub. Keep Austin weird, eh?)
Now that you know about the other golden ratio, here’s what you need to do: Organize your liquor cabinet into two parts, hard alcohol and sweet mixers. Then, make sure you keep a couple of sour mixers in your fridge. When you get home on Friday, just grab a bottle from each side of the liquor cabinet and one more from the fridge, pour, and — voila — instant happiness.
Wanna get a little crazy? Find your favorite cube-shaped random number generator, give it three rolls, then choose the appropriate item from each column in the table below.
Die Roll |
Alcohol |
Sour |
Sweet |
1 |
Tequila |
Lemon Juice |
Simple Syrup |
2 |
Vodka |
Lime Juice |
Triple Sec |
3 |
Rum |
Grapefruit Juice |
Cointreau |
4 |
Rye Whiskey |
Strawberry Shrub |
Gran Marnier |
5 |
Bourbon |
Bloody Mary Mix |
Honey-Ginger Syrup |
6 |
Mezcal |
Dry Cider |
Grenadine |
Personally, I’m hoping for 6-3-5, which is kind of like a Mezcal Paloma, sort of like a Honey and Smoke, but not really similar enough to be either one. So, I guess I get to name it. And given what I’ve heard about the fat-burning properties of honey and grapefruit juice, I’m going to call this newly minted beverage the Weight Watcher.
Now we just need to come up with names for the other 215 combinations. I’ll get started on that right away… soon as I finish this drink.
Two the Hard Way (and an Easy Way)
During our trip to Arizona for winter break, two problems surfaced organically while we were on holiday. (Sorry. Two math problems. There were lots of non-math problems, too, but we don’t have time for all that.)
On the plane, the interactive in-flight map showed the outside temp, toggling between Fahrenheit and Celsius. That led to the following MJ4MF original problem, which I thought — and still think — is pretty good:
The conversion between Fahrenheit and Celsius temperatures follows the rule F = 9/5 C + 32. Sometimes, the temperature is positive for both Celsius and Fahrenheit; sometimes, the temperature is negative for both Celsius and Fahrenheit; and other times, Fahrenheit is positive while Celsius is negative. What is the least possible product of the Fahrenheit temperature and its corresponding Celsius temperature?
You can pause here if you’d like to solve this before I present a spoiler.
Before I reveal two different solutions, allow me to digress. It could be that the statement about the temps sometimes being positive and sometimes being negative is denying a teachable moment. The graph below shows the linear relationship between the two temperature scales. Perhaps a good classroom question is:
When will the product CF be positive and when will it be negative?
Or maybe a better question is:
When are C and F both positive, when are they both negative, and when do they have different signs?
So, to the problem that I posed. As I thought about it on the plane, I concluded that if F = 1.8C + 32, then the product CF = 1.8C^{2} + 32C. I then used calculus, found the derivative (CF)’ = 3.6C + 32, set that equal to 0, and concluded that C = ‑8.89, approximately. The corresponding Fahrenheit temperature is F = 16, so the minimum product is roughly ‑142.22.
Using calculus was like rolling a pie crust with a steamroller, though. I could have just as easily graphed the parabola and noted its vertex:
If I had used the other form of the rule, namely C = 5/9 (F ‑ 32), things might have been a little easier. Maybe. In that case, setting the derivative equal to 0 yields F = 16, which is arguably a nicer number. But then you still have to find the corresponding Celsius temperature, which is C = ‑8.89, and the product is still roughly ‑142.22. So, not much easier, if at all, and again graphing the parabola and noting its vertex would have done the trick:
The only real benefit to using this alternate version of the rule is that it provides a reasonable check. Since both methods — and both graphs — yield an answer of ‑142.22, we can feel confident in the result.
But there’s an easier way to solve this one.
Thinking this was a good problem — and because I like when my sons make me feel stupid — I gave it to Eli and Alex. Within seconds, Eli said, “Well, F is positive and C is negative between 0°F and 32°F, so the minimum will occur halfway between them at F = 16. That means C = ‑80/9, so it’s whatever ‑1280/9 reduces to.” (Turns out, -1280/9 = ‑142 2/9 ≈ ‑142.22.)
Eli hasn’t taken calculus, so he doesn’t know — or, at least, he hasn’t learned — that the minimum product should occur halfway between the x– and y‑intercepts of the linear graph. Yet, he had an intuitive insight that just happens to be true. As a result, what took me about five minutes of deriving and manipulating took him about five seconds.
The second problem arose at the grocery store. Among our purchases was a box of sugar cubes, which contained, surprisingly, 126 cubes.This number is surprising in the sense that it’s not a number you’ll see very often, except for an occasional appearance in the ninth row of Pascal’s Triangle, or maybe if you’re a chemist searching for stable atoms.
A question that could have arisen from this situation involves surface area and volume:
A rectangular prism with integer dimensions has a volume of 126 cubic units. What is the least possible surface area?
That’s not the question that was shared with Alex and Eli, though. (The answer, if you care, is 162 square units, which results from a 3 × 6 × 7 arrangement — which, in fact, is the exact arrangement of cubes in the box above. I suspect this is not a coincidence.)
The problem that I shared with my sons involved probability:
Imagine that the arrangement of cubes is removed from the box intact, and all six faces of the prism are painted red. Then one of the sugar cubes is selected at random and rolled. What is the probability that the top face of the rolled cube will be red?
The boys made an organized list, as follows:
Painted Faces | Number of Cubes |
3 | 8 |
2 | 40 |
1 | 58 |
0 | 20 |
Further, the boys reasoned:
- P(cube with 3 red faces, red face lands on top) = 8/126 x 1/2 = 8/252
- P(cube with 2 red faces, red face lands on top) = 40/126 x 1/3 = 40/378
- P(cube with 1 red face, red face lands on top) = 58/126 x 1/6 = 58/756
Therefore,
- (P of getting a red face) = 8/252 + 40/378 + 58/576 = (24 + 80 + 58) / 576 = 162/576 = 9/42
Wow! That seems like a lot of work to get to the answer. Surely there’s an easier way, right?
Indeed, there is.
Notice that the penultimate step yielded the fraction 162/576. The numerator, 162, may look familiar. It’s the answer to the question that wasn’t asked above, the one about the least possible surface area of the prism. That’s no coincidence. In total, there will be 162 faces painted red. And there are 6 × 126 = 576 total faces on all of the sugar cubes (that is, six faces on each cube). This again suggests that the probability of rolling a red face is 162/576.
Did you happen to notice that the volume and surface area use the same digits in a different order? Cool.
So there you have it, two problems, each with two solutions, one easy and one hard. Or as mathematicians might say, one elegant and one common.
It’s typical for problems, especially problems worth solving, to have more than one solution strategy. What’s the trick to finding the elegant solution? Sadly, no such trick exists. Becoming a better problem solver is just like everything else in life; your skills improve with practice and experience. It’s akin to Peter Sagal’s advice in The Incomplete Book of Running, where he says, “You want to be a writer? […] Just sit down and write. The more you write, the better a writer you will become. You want to be a runner? Run when you can and where you can. Increase your mileage gradually, and your body will respond and you’ll find yourself running farther and faster than you ever thought possible.” You want to be a problem solver? Then spend your time solving problems. That’s the only way to increase the likelihood that you’ll occasionally stumble on an easy, elegant solution.
And every once in a while, you may even solve a problem faster than your kids.