Burning off some anger at well meaning but objectionable FB people

Another one of these things is floating around facebook and watching the commentary about it made me angry. SO here we are. The original looks something this: 

9 = 90
8 = 72
7 = 56
6 = 42
 
3 = ?

And as typical, there are a variety of answers, and as usual, most of them seem to make 'sense' to the commentator, but are untenable in any respectable arithmetic system, which is one in which the following hold

1. we're working in base 10, and not hexadecimal or something (I can't do bases in my head fast enough to know if, given the equations we know, the answer would be different in hexadecimal or something)
2. arithmetic operations are vaguely consistent
3. that the operator/symbol = means 'is equivalent to'
4. arithmetic operations respect RST, that is Reflexivity (that α must be equivalent  to α [i.e. itself]), Symmetry (that α+β must be equivalent to b+α), and Transitivity (that if α is equivalent to β and β is equivalent to γ then α must be equivalent to γ)

In such a system the answer is (and pretty much can only be) 12. 

Now I'm happy to discuss possible arithmetic systems until the cows come home. I'm happy to discuss possible operations. I'm happy to disagree about the right answer. What made me angry was the people who were trying to be 'reasonable'. "Well," they suggest, "the answer is 12, but I suppose the answer could be ... if ..." But it couldn't be. Here's why.

First, a word on notation. There is no sense in which the expression 9 = 90 makes any sense, in a system where the = operator means 'is equivalent to '. If 9 = 90, then what does = mean? If = means 'is equivalent to' we can assume that there is some unexpressed function being applied to the numbers on the left, resulting in the numbers on the right. Let's use the notation f(x) for this operation, so the list should read f(9)=90, f(8)=72, etc. So let's assume the best of intent, that = means =, that RST hold, and so on. 

Given the four examples we know the right answers for, it would appear that f(x) is represents the operation x*(x+1). In which case f(3) is clearly 12. f(x) could only yield the equations it does, i.e. f(9)=90, f(7)=56, etc. and f(3)=18 if the f(x) operation was actually x*(some number above it in the list). This is not what I would accept as a coherent function in any reasonable arithmetic system.

But arithmetic operations cannot be defined in terms of what's around them, or any circumstantioal information.  If f(x) were anything like the above, then changing the order or spatial arrangement would yield idiotic results:

f(9) = 90
f(6) = 54
f(7) = 24
f(8) = 56

But if that were so, we'd have violated RST because f(6) cannot be dependend on to equal f(6). It makes no sense to refer to circumstances in the definition of arithmetic functions. You can't define + in terms of what's around it, or where it is on the page, or even whether you're adding apples or oranges. α+β cannot equal one thing at the top of a page and something else in the middle, any more than it can (reasonably) equal one thing on Tuesday, and something else on Thursday, unless it's a leap year, unless the leap year ends in 00.*

*I suppose there is a case to be made that operations could refer to intrinsic properties of its operands, i.e. f(x) is a different operation if x is even vs odd, or positive vs negative. But that involves universal properties of numbers (in the system in question), i.e. something which are deterministic and 'available' by virtue of the number system you're using, and totally independent of the context in which the number appears**

**I'm not sure this would actually make sense, but I asked someone earlier how you multiply or divide Roman numerals.  I mean, I get that your average Roman kid probably didn't have arithmetic homework in which they regularly had to multiple 46 by 331 or something.  But someone must have had to work out how many denarii to charge someone buying XLII eggs and III denarii a pop. The vest answer I could find needed to refer to whether or not the operands (or their derivatives in the procedure) were even or odd (in the sense that the procedure described halving some numbers and if the number was odd, they had to remember to deal with the remainder.  But I digress.

And even if we did allow that kind of variability, there's still no way to interpret both f(9) at the top of the list and f(8) in the middle of the list, since there is no definable object to use with the f(9). 

If we decide to allow that like the blank space above the top of the list allows for some other, unknown, object to be part of f(x) at the top of the list, then I want to know whether than exception to f(x) also applies in the case of f(3), since there is equally a blank space above the f(3) line and the f(6) line. Which would still be inconsistent with f(3)=18 (f(3) would equal 30, I guess).  In fact f(3) would have no determinable answer, and the arithmetic system being described would fall apart; RST would not and could not apply.***

***I haven't thought seriously about RST and they means in an arithmetic system since high school. Long before my present understanding of number theory set in. This has actually been kind of fun, in the way that potentially learning how to multiply and divide Roman numerals was fun. But I'm still angry at these "well, I guess it could be 18..." people.

So the answer has to be f(3)=12 give my assumptions, and anything else results in an inconsistent and untenable arithmetic. Thus spaketh Hagiwara. Go in peace.

Rob's incredibly satisfying, and not at all psychotic, approach to cutting oblong sandwiches

 I've decided to record for internet-mediated posterity certain snippets of brilliance that, since I have no one to pass them on to, would otherwise be consigned to the void upon my passing. 

This is something I started doing in grad school, when I satisfied certain nutritional and culinary impulse by baking half-sheet (or large baking sheet) sized loaves of focaccia.  Rob's focaccia involves 2.5 cups or so of flour, a couple teaspoons of active dry yeast, about a cup of warmish water (or some combination of water or milk), and something on the order of half a cup of olive oil, and a pinch of salt.  Stir to combine, using just about any method you choose to (dump it all together all at once, bloom the yeast in the warmish liquid with or without a pinch of sugar or some flour, go crazy).  When it starts to come together go ahead an knead it lightly, again in any way you prefer (10 or so minutes by hand, 4-5 minutes in a mixer on low speed with a dough hook, whatevs).  When it comes together into a softish, not sticky, dough, let it rise, gently covered, in a heavily oiled bowl, until it about doubles, which depending on your yeast, your kneading, the temperature you're doing it and so on, might to 45 minutes to 90 minutes.  If it's not doing anything at 45 minutes, bloom a little more yeast in a little more water/milk (warm or cold at this point), and stir it in, re-knead with a little more flour until it looks like dough again, and put it back in an oiled bowl and let it rise..  If it still doesn't do anything, your yeast is dead. 

Turn your risen/proved dough onto a greased baking tray (1/2 sheet pan or a jelly-roll pan, or something about the size of your oven) and press it out with your hands.  You're going for a combination of pressing/rolling kinds of motion and stretching (like a pizza dough) action.  It will fight back so when it doesn't want to spread out any more without bouncing back a lot, cover it with whatever you covered it with when it was rising for a few minutes (to let it relax) and then continue, until you have an oblong, vagule rectangular, flattish 'loaf' of dough.  poke your fingertips into the surface all over, to get a well dimpled surface.  Cover again and let it rise for about half the time of your first rise, when it should expand a little and rise to at least an inch to about 2 -2.5 inches thick. 

While it's rising, preheat your oven to 325-350.  

Uncover your tray and drizzle olive oil over the dimpled surface of the loaf.  Use a pastry brush or gentle fingers to brush some of the oil over the surface so it's all covered, but you still have some shallow puddles in the dimples. Sprinkle the top with a little sea salt, garlic or garlic powder, pepper, herbs, etc to taste.

Place on middle rack of oven, uncovered., and bake for 20-30 minutes or so (depending on your focaccia's hydration, oil content, temp and conditions in your oven, and so on.)  When it's golden brown over most of the surface, it's probably done.

Remove from the oven and let it cool to room temp on a rack. Or in the tray. Whatever. Cope.   

When it's cool enough to touch it can be served. I might slice it into thin breadsticky things for dipping.  Or cut it into cubes for using as croutons (as is or with an extra crispy-crouton step).  For sandwiches, cut the thing into four quarters and have four part-loves to make into sticks, or croutons, or make into sandwiches. (If you're feeding party, make one huge sammich and slice it into 2-3 inch mini sammiches.)  For each sandwich, slice the loaf into top and bottom slices.  Like you would a layer cake.  I don' t know how else to describe it. Don't just build your sandwich with one piece ont he bottom and another on top.  Use one piece to make both the top and bottom of your sandwich.  I don't believe I have to write that, but I'm sure someone won't have the common sense (or food-tv-watching experience) to do it right and I don't want to take the blame for the result.

My favorite treatment for the sandwich is sliced turkey, with pesto. If straight pesto is too strong for you, thin it with olive oil or stir some mayo into it. Good with or without cheese (provolone is a favorite, but anything you like will do--or just skip it. Up to you.)  I like to add a layer of leaf lettuce (green or red leaf lettis, bibb or butter lettuce leaves I use more or less whole, romaine I usually cut into slices, iceberg I shred)

You now have a large, filling, vaguely oblong sandwich that might be anywhere from 7-10 inches long or more, and 5-8 inches wide.  Not precisely wieldy if you like to pick up your sandwiches whole when eating them.  So you have to cut it into pieces to make it wieldy.

This is where Rob's incredibly satisfying, and not at all psychotic, approach to cutting oblong sandwiches comes in.  Halves are stlll a little unwieldy (and for larger sandwiches, half is not quite enough for a satisfying meal even if the whole thing is a little too much, and quarters are a little small to feel satisfying in the hands when eating, even if that gives you 3/4 to be a satisfying meal with 1/4 left over for a little snack later).  So you want to cut the thing into thirds.  But as the thing is oblong, with straight cuts on two sides, and only one right angle, it' s not obvious how to create vaguely even-in-volume and satisfactory edgedness.  So do what I do.

Make one cut (1 on the diagram) across the cut short edge of your sandwich. You now have a trapezoidal piece with only one, short, 'crust' edge and three cut edges, about 1/3 of your original sandwich. Not satisfying from the edgedness front, but big enough to be held in the hand satisfyingly and with enough corners and narrow edges to make eating pleasurable.

Then make a not quite perpendicular cut across the larger piece, roughly from something like where the fourth corner would be to  not quite the middle of the first cut (2).  This produces two portions with fairly long crust edges, and satisfyingly pointy corners with at least one cut side.

The result is three very different shaped sub-sandwich portions, but of more or less equal apparent volume, and each with its own unique and satisfying angles of approach.  Seriously.  Satisfying, and not at all psychotic.

If not eating immediately, wrap in parchment or waxed paper or 'basket liner', and (if your sandwich ends up being the right size) slip into a zip-lock or a sandwich holder or a lunchbox..  Keep cool until you're ready to eat.

Unwrap as necessary, select a portion, and shove it in your face in whatever manner you find most appropriate (and satisfying) given your face and which piece you're working with. Presumably in mannerly and satisfying, and not at all psychotic, bites.