How Many Colored Tetrominoes?

Question: How many different colored tetrominoes are there if we allow only four colors total?

Second question: What the heck is a tetromino?

Dominoes are a great game with rectangle tiles, composed of two adjacent squares with certain numbers of pips on them. A tetromino is a group of four adjacent squares, each sharing at least one side with at least one other square. In other words, those little falling shapes made popular in the game Tetris, and all of its knock-off variations, as seen below:

There are five basic arrangements, if you allow for reflection. (That is, if you allow for picking a piece up and flipping it over.) If you only allow for rotation, then there are seven shapes, each of which can be designated a letter of the alphabet to describe it.

In most games, the shapes are different colors because a) it's a great visual, and b) it's a clue to the player that, say, a "J" is falling not an "L". Ditto for the "S" and "Z" pieces.

As with any successful game, there have been many imitations and variations. Even games that are somewhat unrelated produce their own variations, which are suddenly similar to Tetris. I've seen a few of these where the pieces, for a multitude of reasons, are multicolored instead of monocolored, as shown below:

This lead me to thinking about the number of possible colored blocks that could fall in the game of varying shapes and color schemes. My only arbitrary limit was that each block had to contain each of the same four colors. (Naturally, I picked red, yellow, green, and blue, pretty much by default.)

First instinct is to use the Counting Principle: the number of shapes times the number of four-color arrangements. That would give us (7) X (4!), or 7 X 24 = 168.

Unfortunately, first instincts may put you on the right track, but they sometimes leave things out. In this case, we can't forget about rotational symmetry.

The I piece (the line) has rotational symmetry of Order 2. If you rotate it 180 degrees, it looks the same. That makes it the same twice in one 360-degrees rotation. The S and Z pieces also have Order 2 symmetry. The O piece (the square) has rotational symmetry of Order 4. If you rotate it 90 degrees, it looks the same. That makes it the same four times in one 360-degrees rotation.

So we have 3 shapes with 4! color variations, 3 with 4!/2 variations and 1 with 4!/4 variations: 3 X 24 + 3 X 12 + 1 X 6 = 72 + 36 + 6 = 114 possible tetrominoes.

For some reason, I feel better knowing this, and maybe it won't distract me so much next time I play one of these games on the train where I have no wifi. (A Tetris Offensive, perhaps?)

(As always, you're free to correct my math. In the event of an actual mistake, I'll edit my work and pretend I have no idea what you're talking about.)

N-RN.3 (Real Number System) - Irrational Behavior

A short entry for tonight.

Moving on to another Common Core Algebra standard brings me to N-RN.3, which reads

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

The key word here is "explain". Before we do that, let's establish that the rest of the standard is true: if you add two rational numbers, you will get an irrational number; the sum or product of a rational and irrational number is irrational. (Leaving out the obvious case of multiplying by zero.)

If we remember that the word "rational" comes from "ratio" and that ratios can be written as fractions with numerators and denominators that are integers. Then the sum of any two rational numbers can be written as the sum of two fractions. All we need to add these two fractions is a Common Denominator, which can be obtained by multiplying the denominators. The set of integers is closed under multiplication; the product will be another integer. If we add the numerators, the set of integers is also closed under addition. The total will be another ratio of two integers, which must be a rational number.

To show that the sum of a rational number and an irrational number is irrational, we can use contradiction. Supposed that r is rational and x is irrational. (Let's not use i as it has a different meaning, which may cause confusion.) Assume there is a sum r + x that is rational. If we add -r, which is also rational, we get r + x - r, or just x, which must be rational because the sum of two rational expressions is rational. This is a contradiction because we started with x as irrational. So the sum of a rational number and an irrational number must be irrational.

The same contradiction can be used to show that rx cannot be rational if x is irrational, by dividing both sides by r.

Now that we have that out of the way, the fortunate thing for Algebra students is knowing when the product or sum is rational or irrational. Don't be quick to assume that a radical sign indicates irrational. They love using the square root of 64 or the cube root of 27, both of which are perfectly rational. However, it helps if you can explain why it is so other than to say, "Well, you know, it's obvious." (And, hopefully, it is.)

What Makes the Golden Ratio so Golden?

Now that I’ve concluded my Golden Ratio-themed comic serial, I wanted to get into a little what the Golden Ratio actually is. Just saying that it’s some number that’s approximately 1.618, doesn’t quite do it justice. What’s so special about that number? And what makes that a ratio?

Second point, first. A ratio is a comparison of two numbers, so it can cause confusion if only a single number is written, even if “to 1” is implied.

What makes it so special is what is being compared. Look at these two images, for example. They are basically the same thing, only the notation for the lengths are different.

Suppose we wanted to find a ratio of the length of the square to the length of the rectangle, and we wanted that ratio to equal the ratio of the length of the smaller rectangle to that of the bigger one, what value of x would accomplish that?

It terms out that we could set up the problem in either of these two ways (and many more, besides!). On the left, the square has a length of 1, the smaller rectangle has a width of x, so the bigger rectangle has a total length of x + 1. One the right, the square still has a length of 1, but the bigger rectangle has a total length of x, so the smaller rectangle has a length of x – 1.

Setting up the proportions and cross-multiplying we find:

In either case, we get the same quadratic equation: x² – x – 1 = 0.
That’s not something easily factorable, so we have to use Ye Olde Quadratic Formula, after which we discover that the roots are

If we take the positive value, we get x = 1.618033988749894848204586... But what about the negative value? If we subtract root 5 from 1 and divide by 2, we get x = -0.618033988749894848204586...

Look at the decimal portion. They are the same! Keep in mind, that we’re dealing with irrational numbers here, which aren’t supposed to conform to patterns, but this one is just brilliant. And, yes, there is a reason for it.

Take a look at that second proportion, above, on the right, with the removable 1 in the denominator.

This is saying that one less than the number is the reciprocal of that number! If you divide 1 by 1.618033988749894848204586..., you will get 0.618033988749894848204586..., the same as if you just subtracted one.

This works for the negative value as well: If you divide 1 by -0.618033988749894848204586..., you will get -1.618033988749894848204586..., the same as if you just subtracted one.

One last point, and a hat tip to William Ricker for mentioning it, how else can we write the reciprocal of x, 1/x? What exponent gives us the reciprocal of x? An exponent of -1. That means that this equation, this proportion, can be written as:

Absolutely brilliant. And quite Golden, if I say so myself.

A Little Bit of Fibonacci ...

Today's column isn't going to happen the way I wanted it to, so I'll try it for tomorrow. In the meantime, I'll leave you with this:

I just finished posted a series of comics with crazy numbers in the titles after the word "part", instead of the normal "Part 1/10, Part 2/10, ..." etc.

This is because those numbers were intended to be ratios, and not just any ratios, but Fibonacci number ratios. As you might have guessed from the first episode in that sequence, I would be working with Fibonacci numbers. You might also have guessed that I would have been done by the time I got to 21 -- That's okay! I was expecting to be done a lot sooner, too!

As I hope to get into a little more depth tomorrow, when you divide two consecutive Fibonacci numbers, you get a ratio that is close to the Golden Ratio, which is also known as "phi", and is approximately 1.6180339887. The larger the terms, the smaller the difference between the two ratios.

Here is a table showing you the first 15 terms:

But that's not the only thing interesting about the Golden Ratio. (Well, of course not! Books could -- and have -- been written!) There should be more about it in tomorrow's entry.

Math Isn't Everything. It Isn't Even the Only Thing.

Math isn't everything. It isn't even the only thing.

That may sound shocking coming from me, but theoretical math (you know, a lot of that Algebra stuff) needs to be applied to real life using real life conditions, which sometimes (most times?) take you out of the scope of any classroom problem. This is why some people don't think they're using algebra (when, in fact, they are), or why they think it isn't really practical.

A quick example of what I mean: there's a joke floating around that goes something like this: only in a math class can you buy 36 oranges and 25 apples and not be thought crazy.

Here's a different example (not a joke). Each weekend in August, the Miller family barbecues six hot dogs. Buns come in packages of eight. Over four weekends, how many packages of buns should they buy?

Don't scroll down until you're ready for the answer.

The "correct" answer is four packages.

Now, wait a minute, you protest! The cook 24 franks, they need 24 buns, and you get 24 buns in three packages of eight.

That is certainly true. On the second weekend, you still have 2 buns leftover from the prior weekend. This brings the next question -- the real world question -- who gets the stale buns? Probably the shy, quiet one who doesn't speak up for himself. Or the youngest one who doesn't know any better. Most likely, Mom, who who sacrifice for her children, giving them the food from her mouth if need be, assuming she wanted two hot dogs to begin with. (Take better care of Mom, she's been good to you!)

In the real world, even if the bread hasn't reached it's expiration date, those leftovers still won't be as fresh as new rolls will be. Moreover, consider the fourth weekend. All of the bread is leftover, and no one gets a fresh roll. If you're not on a really tight budget, buy new bread each week.

What do you do with the extra bread? Feed the birds. Make breadcrumbs. Have a really funky looking sandwich on Monday.

What do I know? I'm a math teacher, not a cook.

N-RN.2 (Real Number System) - Rationalizing the Denominator

This is the third and final column on Common Core Standard N-RN.2. If you missed the first two parts, Part 1 dealt with evaluating expressions with rational exponents, and Part 2 showed how to simplify using factor trees and how to add and subtract radicals. The last piece of this standard (and since I'm only dealing with part ".2", I could really call it a "substandard" if I wanted to, mockingly) is to "Simplify radical expressions by rationalizing the denominator (Algebra 1 - EE.2)".

Previously, we mentioned that you can multiply two radical numbers by multiplying their radicands. We also factored radical numbers in order to simplify them. Let's talk about division. When you divide, you multiply by the reciprocal; that is, you can create a fraction of the two numbers without relying on your early education "gazintas". (You remember, "2 gazinta 6 three times".) Likewise, when you take the square root of a fraction, you are actually dividing one radical number by another.

So if you wanted the square root of 1/4, you would take the square root of the numerator (radical 1 is 1) over the square root of the denominator (radical 4 is 2). The result would be 1/2.

But suppose we wanted the square root of 1/2? Again, we can split it up into the square root of 1 (which is 1) over the square root of 2.

Here's where we run into a problem because there are rules from fractions. One of them is that there cannot be any radicals in denominator. You have to get rid of them.

We haven't discussed this before, but there's really only one simply way to get rid of a square root sign: square the number. We need to multiply the denominator by radical 2. We are allowed to do this because it's a fraction and we won't change the value of the fraction at all as long as we multiply the numerator by the same amount as the denominator. The fractions (square root of 2 over square root of 2 looks scary to evaluate until you remember that any number, even an irrational, divided by itself is one, with the exception of zero. If you multiply a fraction by 1, it doesn't change its value, even if it looks different. The result is that the radical is gone from the denominator and has moved into the numerator, which is allowed.

One more example. Try it yourself before scrolling down and looking at the image. What is the square root of 4/5?
Take the square root of each number. Rationalize the denominator. What's left in the numerator? What's left in the denominator?

Okay, check your work.

That's it for this standard. Time to move on to part 3, coming soon.

N-RN.2 (Real Number System) - Dealing With Radicals

In a recent post, we explored evaluating expressions with rational exponents in them, but there's more to Common Core Standard N-RN.2. Don't worry, some of it's easier to deal with than what we already tackled.

These are the items listed below the standard, at least according to the IXL website, where I found the list:

• Simplify radical expressions (Algebra 1 - EE.1)
• Simplify radical expressions by rationalizing the denominator (Algebra 1 - EE.2)
• Multiply radical expressions (Algebra 1 - EE.3)
• Add and subtract radical expressions (Algebra 1 - EE.4)
• Simplify radical expressions using the distributive property (Algebra 1 - EE.5)

Is there anything else to deal with? I don't know. Dealing with rational exponents isn't in this list, and yet I think that they might be encountered before Algebra 2. FYI, the notation "EE" stands for Expressions and Equations, which allows me to once again state that "Expressions don't have equal signs and are evaluated, and Equations do have equal signs and are solved."

Simplifying radical expressions is not a difficult task -- as long as you know that it does NOT mean pushing buttons on your calculator and coming up with an approximate decimal equivalent to 8 or 12 or 15 decimal places. Simplifying a radical is similar to reducing a fraction to its lowest terms. It makes it easier to deal with for computations (particularly adding and subtracting, when the radicals have to be "like terms") and comparisons. If the only thing you're planning to do with a radical number is square it, then, yes, simplifying it is a bigger waste of time than converting an improper fraction into a mixed number when it's only going to be used for slope.

There is a very straightforward method of simplifying square roots, but it seems to mystify some of my students who, apparently, never grasped the concept of what a square root (or a perfect square) was in the first place. They memorize steps, but uncertainty about the order causes them to mess up at the very end, removing radical signs from irrational numbers or leaving them in after taking a square root. (For example, they'll write that the square root of nine = the square root of three, instead of three.)

The simplest method involves finding the largest perfect square which is a factor of the radicand (i.e., the number under the radical sign). If it isn't the largest perfect square, then the radical hasn't been fully simplified. An example:

One problem my students face is not understanding the concept of a perfect square, so instead of 25 and 2, then use 5 and 10. After that, they're stuck, or they just decide, for example, that the square root of 5 is the same as 5 without the radical sign.

Because of this, I tried a different approach, using factor trees. They remembered doing them in middle school, and actually liked using them again. (You see, your teacher was right! You are using them again!) The example looked something like this instead:

After they have the prime factorization under the radical, I have them circle the pairs of numbers, cross them out and write one factor outside the radical. This has two downsides to it: first, if the number has a lot of factors, there will be a lot of extra work (but at least they will know, for certain, that they simplified their answer); second, if they don't complete the problem, they basically just drew a factor tree, which looks kinds childish and silly from a high school student.

Multiplying, Adding and Subtracting Radicals

Multiplying two radicals is as simple as multiplying two fractions. Just multiply the numbers under the radicand. For instance, radical 7 times radical 10 equals radical 70. If the number can be simplified, do it, according to the rules above. Obviously, if you square a radical, such as radical 6 times radical 6, the radical symbol goes away. In this case, you get radical 36, which is just 6.

As mentioned above, if you want to add or subtract radicals, they have to be alike. You can't add or subtract the following the way they are:

They aren't alike. It's like two to add 5² + 4² and getting 9². (In other words, you don't.)

But if you simplify the radicals, how to combine them becomes much clearer:

Finally, there is Division, but I'll save that for another column because of the standard, above, Simplify radical expressions by rationalizing the denominator.

N-RN.2 (Real Number System) - More Rational Exponents

Continuing my post from Tuesday, on the Real Number system and rational exponents, let's move on to Standard N-RN.2, which reads Rewrite expressions involving radicals and rational exponents using the properties of exponents.

There's a lot to consider under this standard, so I'll continue with rational exponents, i.e., fractions. What if we wanted to evaluate an expression like this?

We need to recognize that the radical 5 is the same as 5^1/2, so

The rules for exponents say to multiply the 1/2 and the 4, giving us 5² or 25.

We can take this further. Suppose we had

The cube root is the same as 1/3 power. So

We can evaluate 6³ as 6 * 6 * 6 = 216.

One more example: How would we handle

The fourth root becomes the 1/4 power.

Now we can get a little fancy with and deal with the multiplication of two fractions:

One final note: The answer doesn't always have to be a rational number. You may exchange one rational power for another, one root for a different one. Consider:

Problems could contain any combination of roots and improper fractions, which may or may not have a simple rational answer. But keep the calculator handy just in case you need to know the sixth root of 117,649. Showing your work, of course.

N-RN.1 (Real Number System) - The Meaning of Rational Exponents

While I'm trying to update every day in August, I might as well start taking a closer look at the Common Core standards, which have now been in place for one year in high school Algebra 1 classes in New York. The first standard I find is N-RN.1. The first N stands for Number and Quantity. Unfortunately, the RN stands for Real Number System and not Registered Nurse because the latter would be helpful when you got sick of all this! I could have helped that telling that joke -- I chose to tell it any way, if only because I had to look up what the letters meant, particularly the first "N".

The entire standard reads as follows: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

It then has two components, one for Algebra 1, the other for Algebra 2. The Algebra 1 component is Evaluate integers raised to rational exponents (Algebra 1 - V.9). Don't ask me about the "V.9", I've done enough searching for this column.

I used to start teaching each year with Order of Operations, something that all the students should have seen before, and yet seemed to forget about. They know an acronym, such as PEMDAS, but don't know what it means. Oh, they know what the six letters mean, but they don't get the concept. And even when they can explain the concept, when push comes to shove and the pressures on (and they're taking a quiz), you find them calculating from left to right as if they hadn't learned anything. I had to change that when we started welcoming calculators into the lesson (even before we started requiring them). The calculators were down the work for them, so they didn't have to learn it, right? Wrong. I just adapted the problems. I started added more operations within fractions and adding exponents, forcing them to pay attention to what they put into their calculator. For instance, you need explicit parentheses to group things in a calculator because the numerator and denominator are implicitly grouped.

But Common Core changes that. They have to learn it much earlier, so they're ready for Algebra by the time they get to high school. Yeah, right. I'm still doing it. But wait, there's more.

After parentheses, come the exponents. Some students know the concept of exponents, and some just press buttons on the calculator. They know (or they'll learn!) that they are multiplying the factor some number of times. What they haven't seen before is a fraction as an exponent. What do you do when you see a fraction? Hide under the desk, usually. Wait for it to be over. I'm not exaggerating much.

First, I have to teach the concept that exponent 1/2 means take the square root, as stated in the gem. And then see how well they know they're square roots, either with or without a calculator. (I try to get them to learn up to 256, with some success.)

Second, I have to teach them that exponent 1/3 means take the cube root. This usually entails explaining what a cube root is, and where to find it on the calculator. Maybe reviewing what "cubing" means, and possibly through in Volume = length X width X height somewhere, just so I can spiral back to it when that comes around again. Depending on the results, I could try to conquer the concepts of fourth and fifth roots. Seriously, a practice webpage asked me (729)^(1/6).

At this point, I'm asking: are they getting it, or are they pushing buttons? I don't mind the button pushing if they understand the concept, because then they'll start recognizing patterns and will be smarter about their button pushing.

Okay, so the next step is the real doozy: explaining exponents of 2/3 or 3/5. The students have to deconstruct the fraction. That is, they need to know that 2/3 = (2)X(1/3), so they need both to square it and then take the cube root. And then I'll suggest that you take the cube root first, so that they're dealing with smaller numbers. Sometimes this makes sense to them.

Finally, there's the kicker: improper fractions. If fractions are Dr. Frankenstein, improper fractions are his monster. They're a whole new level of scary, and the first thing they want to do is turn them into mixed numbers. Or decimals. No! Wait! Stop!

Taking the (5/2) power of 16 is as simple as taking the fifth power of the square root of 16. Okay, read that again like you're a ninth grader.

No, it's not that difficult to do once the concept is learned, but it's something they haven't seen before, and they're learning it earlier. I can't remember exactly when the first time I was required to find a third or fourth root. I was probably using logs to do it. But then, I didn't have the calculators that they have today on their phones. They'll have the answers at their fingertips, but I'll make them show the work so I know they get the idea behind it.

Hey, Internet: Where's My Picture of Me and Ann B. Davis?

The news of the death of actress Ann B. Davis was a bit of a shock. For those of a certain age, she was "Schultzy" on the The Bob Cummings Show. I'm NOT that age, having grown up instead in the golden age of first-run episodes of The Brady Bunch. Totally tangential, and not to make light of her passing because it does sadden me, but a burning question has been brought once again to the forefront of my noggin from the deep recesses where it had been locked away for many years:

Where is my picture of me and Ann B. Davis

There's a short story here, but I'll be quick about it:

Sometime back in 1993 -- Who am I kidding: it was May 11! I have the ticket stub right next to me! -- I won tickets to a Brady Bunch Reunion Cruise, sponsored by WPLJ-FM radio (95.5 FM, NYC) and The Spirit of New York. The announced guests on the cruise were Barry Williams, Susan Olsen and Ann B. Davis, aka "Greg", "Cindy" and "Alice", respectively, if you grew up on a different planet.

It was a little of an odd evening for me. My wife couldn't make the cruise, and as I would be traveling home late by subway, I didn't want to ask anyone I'd feel obligated to take home at that hour of the night. Especially if there might be by alcohol involved.

I briefly contemplated asking some young lady standing around waiting to get a glimpse of "Greg" by the gangway if she wanted to go dancing, but I found two problems with this. First, there was no guarantee I wouldn't be deserted the moment she got on board (or five moments after we would attempt polite dinner conversation). Second, they keep the gawkers far away from the ship out by South Street and well away from the pier. (Pier 11, if I remember correctly, down by Wall Street.)

Getting back to the story, I was on line alone, listening in to other people's conversations as we waiting to get our passes and board. I finally got mine, except it was someone else's -- they just checked my name of a list and gave me the next ticket. A photographer waited for each couple to start up the gangway to snap a souvenir picture. He asked the couple ahead of me, "Are you three together?" I nodded no, but the woman asked, "Would you like to be together?"

An interesting offer, but I declined, and I had a picture taken on my own. I don't remember if they'd waited for me, or we were just assigned seats for dinner, but the three of us shared a table for dinner. The "couple" turned out to be a mother and son. The guy's name was Dean, and he was around my age. She was Marcy; I don't know her age, but she looked like she'd been a young mother. Not that there's anything wrong with that. Dean and I each thought the other looked familiar. Having both grown up in Brooklyn with two million other people, it was possible. However, we only came up with one mutual acquaintance, and we couldn't think of a time we'd been together with him. (And it was only an acquaintance of mine, not a close friend or anything.)

Soooo ..... BRADY BUNCH! CRUISE!

The Spirit of New York is a nice dinner cruise ship, sailing from South Street along the East River, around the tip of Manhattan and into the Hudson River. The crew was friendly and professional. There were hors d'oeuvres to be eaten, drinks to be drunk, and views to be taken in. The ship sailed, and we went on deck to feel the sea breeze in our faces as we sang The Brady Bunch theme song with total strangers and a drag queen with a microphone as a second one -- a brunette wearing a green dress and carrying a videocamera the size of a small Buick on his (her?) shoulder -- filmed the merriment. They walked about the entire ship the entire evening, at one point being chased up from below decks. Honest mistake.

After dinner, the dancing started on one deck and the celebrities were signing autographs on the deck below. The line never got any shorter. Not until we got on it. A couple of girls got on line behind us and then that was in for the following hour of the cruise. I could've stayed on the dance floor and possibly Electric Slided (electically slid?) into newswoman Naomi DiClemente and had essentially the same place in line an hour later. But conversation with strangers is something New Yorkers do best. That is, when we're not totally ignoring total strangers, which we're pretty good at, too.

So I didn't have a camera on me (or if I did, it wasn't working), but I did have a journal on me. Back then, I had a mini-notebook on me all the time, and I tried to write in it every day. Usually, I was writing while riding home on the subway. I'm sure my poor penmanship suffered, but it's when I had the most solitude to write -- on crowded, evening rush hour subway trains. I worked far enough uptown that I usually had a seat, except when I had to give up a seat for a mother-to-be or the elderly. (Watch out for the Wednesday matinee crowd!)

When we finally got to meet them, Ann B. Davis, Alice, was at the first table. I asked her to sign my journal, opening it to a fresh page. She was impressed that I had a journal. At that point, Marcy asked if I wanted a picture with Ann. C'mon now: Who could say "no" to a picture with Ann B. Davis? Ann was obviously used to this, and had probably posed for dozens of pictures already that night. There's a table between us, so I leaned back and Ann leaned forward. Apparently, we weren't close enough because Ann pulled me back to narrow the gap as Marcy took the picture. (Or maybe she told Dean to take the picture? Could be.)

We met Barry and Susan. Marcy points out all the blank pages in the journal to "Cindy" and says that she needs to write her life story. Susan declines and adds a note in my journal that she's already written her life story. (Thanks, Marcy -- I had a thing for Cindy when I was, like 12 -- go and ruin it for me. Well, there was still Naomi ...)

The rest of the evening was fun ... and short because we really were on line a long time. When we pulled back into port, Marcy insisted that they give me a lift home, from lower Manhattan all the way out to Bensonhurst, my first apartment after I got married. I gave them my address, and they promised to send me a copy of the picture. Well, it never came. It's been twenty years, and still nothing. Granted, I moved out of that apartment within three years, so maybe it's sitting in a Dead letter pile at Bath Beach Station.

However, in the intervening years, the Internet has evolved, and who knows, maybe this will go viral and someone will see it. Maybe that someone will know a Dean who has a mother named Marcy. Maybe Marcy still has that picture of some oddball they met on a cruise, sitting someone in a shoebox with other pictures in the bottom of a closet behind from old mixtapes. Maybe the Internet can finally answer the question for me:

Where is my picture of me and Ann B. Davis?