Fat Rejection Update: October edition

Thanks to everyone for your support and bugged-out eyeball facial expressions from seeing my interim results on Facebook. In some ways getting here was harder than I thought and in some ways it was easier. I haven't flawlessly executed on either my diet or Power 90, but I haven't let slip-ups and flaws prevent me from continuing, and that's been the most important part.

There are some big challenges up ahead. Halloween is the start of the Winter Fattening Season (ending around New Year's). Time for some reminders and motivations.

Motivations

1. Never be fat again
2. Win the love of a beautiful woman
3. Live to 100

Reminders: Rules for awesome eating (see I Can Make You Thin!)

1. If you're hungry, eat!
2. When you eat, eat exactly what your body is asking for
3. When you're eating, eat slowly enough to enjoy every bite
4. If you think you might be full, stop! If it turns out you're still hungry, see #1

These rules are really all you need. You just really need all of them.

Tips by rule

• 4: Clean plate club: LEAVE IT. Here's how you quit: leave something on your next plate. Throw it out, put it in the 'fridge, whatever. Just don't eat it. You don't need it. Leftovers are king. If you follow the 4 rules and go to a restaurant for a meal then you'll likely have a second meal from the leftovers.
• 3: Deep breaths can have the same psychological effect as eating a big forkful of pasta. Try it!
• 1: Drink water. When you really try to follow these 4 rules, sometimes you won't be sure whether you're hungry or not (a big help was to think how full I am on a 1-10 scale)
• 1: Diet pop isn't good for you. No, I don't mean it gives you cancer (to the extent that it does, the heart attack would get you first). No, I don't mean it always tastes awful (e.g. Coke Zero, Cherry Coke Zero, Diet Dr. Pepper are delicious). Diet pop trains you to ignore what your body tells you about when, what and how much to consume. You get the sugary taste without the calories, adding noise to the signals your body relies on for feeling full or craving certain foods.
• 2: On balance, alcohol and caffeine are kind of lame. The Mormons have this one right. Aside from whatever else they do, alcohol and caffeine make it tough to figure out what your body is actually asking for both qualitatively and quantitatively. Don't tell me your body is asking for Costco burritos. I don't believe you.
• 1: Meal times are there to be ignored. After a few days of following the 4 rules, you actually do get into something resembling a breakfast/lunch/dinner scheme, except now it really is good for you. I agree with Scott Adams that "meal time" may be the most dangerous idea in our everyday lives in 21st-century America.
• 3: Different parts of your tongue are sensitive to different tastes. Roll that bite of stuff all over the place and get the full effect--every time.
• 4: Food tastes best when you follow these rules. When your body isn't specifically asking for a certain type of food or if you are full, the food will taste very bland in comparison. This is, in fact, a great way to tell that you might be getting full. I hate the cliché "hunger is the best spice" because to me it implies that if you get way, way hungrier, the food will taste way better, and failing to eat when hungry breaks rule #1.
• 3: Before finding the 4 rules from I Can Make You Thin!, I was trying to count Calories. Calorie-counting is guaranteed to produce results as long as you match your intake and outtake. We're very bad at doing this numerically, which is why a key part of what I've been doing has been listening to my body's signals. That way the four rules don't have to change if you get sick or start working out or get pregnant. Calories let me know about how full a serving of some food will make me relative to how hungry I am. If I'm looking at 200 Calories of candy, not only is it probably not exactly what my body is looking for, it's also going to push back that salmon dinner I was planning to cook a little later.

I'm into the last 30 days of Power 90. This is the home stretch for the program, but I'm only treating the program as a whole as a first step. We're at the start of skiing season in the Pacific Northwest. I'll continue biking to and from work. The Seattle Bouldering Project is open year-round. Now is the time to start looking around for adult sports leagues. I'll continue my thrice-weekly morning martial arts training. I've still got that dog to walk. The end of Power 90 will not be the end of my active lifestyle.


Staying fat is for Santa Claus and Eric Cartman. If you only knew the rolls I've yet to shed and the ferocity with which I'll shed them....

The Joy of Set: Hint 2

Edit: Clearly there's something wrong here: http://www.math.rutgers.edu/~maclagan/papers/set.pdf calculates that you can have 20 Set cards in-play without forming a Set...what's worse, it provides a counterexample to my calculation that you're guaranteed a Set at 14 or more cards.

So what's my mistake below? Simple: a single card may complete more than one Set, and I didn't take that into account. The rest of this post stands as a testament to the human need to simplify, the difficulty of problems in combinatorics and the addictive nature of writing posts with mathematical equations.

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What is the largest number of Set cards that can be in play without having a set present?

We saw in hint 1 that given two Set cards, exactly one card will complete a Set. So for any \(n\)-card Set board containing zero sets, it means we don't have any of the \(\binom{n}{2}\) cards that would complete our set. Since the problem assumes that no Sets have been called, those cards are all hiding in the deck, out of play.

The possible largest \(n\), then, for which we can have n Set cards in play out of the 4³ = 81 without a Set is the greatest integer \(n\) for which \(\binom{n}{2} \le 81 - n\):

\[\binom{n}{2} \le 81 - n\]
\[\frac{n!}{2!(n-2)!} \le 81 - n\]
\[n(n-1) \le 162 - 2n\]
\[n^2 + n - 162 \le 0\]
Testing near the following values of \(n\):
\[-1 \pm \sqrt{649} \over 2\]
Well, really just \(-1 + \sqrt{649} \over 2\), since a negative number of Set cards in play is nonsensical:
\[\frac{-1 + \sqrt{649}}{2} \approx 12.24\]

\(n\) needs to be a whole number. Since \(n^2 + n - 162\) needs to be less than or equal to zero, we'll take the floor for \(n = 12\). Let's check our work:
\[\binom{12}{2} \le 81 - 12\]
\[\frac{12 \cdot 11}{2} \le 69\]
\[66 \le 69\]
checks out. Pushing \(n\) up to 13 doesn't seem promising:
\[\binom{13}{2} \le 81 - 13\]
\[\frac{13 \cdot 12}{2} \le 68\]
\[78 \not{\le} 68\]

Since \(n = 12\) is our last chance to have a board with zero sets, it's tempting to say \(p(n) = 1\) for \(n \ge 13\). This actually isn't the case. Let's look at our interpretation a little closer and go back to the beginning just for a moment:

Since there is exactly one card which will complete a Set from a pair of Set cards, we need all of those \(\binom{n}{2}\) cards to be hiding in the deck. When there are 12 cards in play, the deck has 69 cards in it, and only 66 of those cards are refugee Set-completers. This means that from what's left in the deck, we have three choices for our 13^th card which will not complete the set! The fact that there aren't enough hiding spots in the deck for Set-completing refugees at \(n = 13\) actually means that \(p(14) = 1\). The fact that there are three extra hiding spots for the 66 Set-completers at \(n = 12\) just means that from a board with 12 cards and zero sets we have a \(\frac{3}{66} = \frac{1}{22}\) chance of making it to 13 without a Set. This segues nicely into Hint 3....

\(n\)	\(p(n)\)
0	0
1	0
2	0
3-12	No spoilers!
13	allllmost 1 (not fair, I've already worked out the full answer)
\(\ge 14\)	1

The Joy of Set: Hint 1

Problem: http://gottablogsomewhere.blogspot.com/2012/09/better-living-through-mathematics-set.html

Hint 1: Choosing any two Set cards, there is exactly one card which will complete a Set.

We know this because in a Set, all four attributes of the three cards must be the same, or they must all be different. Each attribute has three values, so the third card has exactly one way to be either the same or different in each attribute from the other two. Each card is unique, giving us 3⁴ = 81 Set cards.