## Posts tagged ‘color’

### Red + Green = Christmas, and 62 Other M&M Color Combinations

‘Tis the holiday season, so every grocery store, pharmacy, and convenience store is now stocking the M&M^{®} Christmas Blend, a joyful combination of red and green button-shaped chocolate candies. It’s unclear whether this mixture actually helps to imbue the holiday spirit, but the consumption of these tasty morsels will make you look just a little more like St. Nick.

As far as I’m concerned, the Christmas Blend — not to be confused with Holiday Mint, which uses a (disgusting) mint chocolate filling — is one of just a few acceptable color combinations. Why? Because it uses colors that can only be found in the original Plain M&M packs, which contain red, orange, yellow, green, blue, and brown.

The original packs didn’t contain white M&M’s — sorry, Freedom Blend (Fourth of July). The original packs didn’t contain pastel colors — hop on by, Easter Blend. And nowhere on God’s green Earth will it ever be acceptable to use white chocolate inside those delectable candy shells — hit the road, Carrot Cake M&M’s. (Yuck.)

As you can tell, I’m a purist, and I have fairly strong opinions about this.

To my knowledge, there are only two other blends produced by Mars, Inc., that satisfy my acceptability criteria:

- Harvest Blend: red, yellow, brown
- Birthday Cake: red, yellow, blue

So, where am I going with all this? Glad you asked.

The Christmas, Harvest, and Birthday Cake blends represent just three of the 63 possible color combinations that can be made from the original six colors. That leaves 60 combinations that are just begging for names.

(A little history. As you may know, I have a quirk. I eat M&M’s in pairs of the same color, so I can place one on each side of my mouth and feel “balanced.” But it’s atypical for a pack to contain an even number of every color. When I near the end, I’m often left with one to six unmatched M&M’s. And I’ve always thought that these various color combinations deserved a name.)

What would you call a combination of red, yellow, and green? Obviously, STOPLIGHT.

What might you call a combination of red, yellow, and blue? Based on the Man of Steel’s outfit, I like SUPERMAN. But Mars, Inc., has already applied the moniker BIRTHDAY CAKE.

What would you call a collection of just green M&M’s? I don’t know — QADDAFI, maybe? (Sorry, dated reference.)

What would you call a combination of orange, green, and brown? I have no idea.

And that’s where you come in.

Below is a Google poll where you can enter a color combination and suggest a name. In early January, for any color combinations that have more than one suggestion, we’ll vote on it. That’s right — crowdsourcing, baby!

But before you scroll and start clicking, let me lay out some ground rules:

- Keep it clean, please, no worse than PG-13.
- No sports teams! Why? Because the Pittsburgh Steelers, Pirates, and Penguins are black and gold… and although yellow is close to gold, there are no black M&M’s in the Plain M&M’s pack, so that combination is not possible. If M&M’s can’t be used to represent my team, then they can’t be used to represent any team. Sorry — my game, my rules. Not to mention, nearly every color combination corresponds to at least one sports team, so it also demonstrates a lack of creativity. Unless, of course, you pick the colors of a team from the Swedish Bandyliiga, but let’s be honest — were you really going to do that?

Some time ago, I tried to craft names for all the combinations on my own, but I failed miserably. You can see how far I got on **this Google sheet**. So you can tell that I really, really need your help.

Have at it, y’all!

If you can’t see the form below, click this link:

**https://goo.gl/forms/jiCEClAMSDTJtHGZ2**

Don’t want to goof around with a Google form? Fine. Place your thoughts in the comments.

### Sock Probability

I don’t know if problems like the following are famous, but there sure are a lot of them online — Cut the Knot, Stack Exchange, and Braingle, for example — and they’re typical for a high school classroom or middle school math competition:

There are 14 red, 6 orange, 10 yellow, 8 green, 4 blue, 12 indigo, and 2 violet socks in my sock drawer. How many socks must I randomly remove from the drawer to guarantee that I have two socks of the same color?

You may or may not know the answer, but the problem itself leads to a follow-up question:

Why the hell do I own socks in every color of the rainbow?

That’s just weird. But if you can get past that, here is a related problem:

If I randomly remove two socks from the drawer, what is the probability that they form a matching pair?

As it turns out, I’m something of a sock aficionado. (Yeah, it’s weird, but surely not surprising. I mean, I write a math jokes blog. You didn’t think I was normal, did you?) Although the context above is fictitious, I am indeed the owner of three pairs of identical socks that look like those shown in the picture. And yes, those are *my* feet and ankles. My mathematical sexy runs all the way down to my toes.

Here’s a close-up of one of them, in case you can’t see it in the larger picture:

That’s a letter **R**, because these socks are specially designed for each foot. The other sock has a letter **L**. (Duh.)

This leads to another mathematical question, more real-world than those above:

I just finished washing these three pairs of socks. While folding them, I selected two socks at random and rolled them together. What’s the probability that there’s one R and one L?

The answer, of course, is **zero**.

Yes, I know that *theoretically* the answer should be **3/5**. But theory doesn’t match practice in this case. When I do my laundry, I sometimes forget to pay attention to the R and the L, and my sock drawer invariably results in one pair of two R’s, one pair of two L’s, and one correctly matched pair. And then when I wake up at 5:30 a.m. and put on my socks in the dark (so as not to rouse my wife from slumber), my feet feel all weird. The one with the wrong sock starts tingling, so I have to remove the socks and choose another pair entirely.

Similarly, here’s another real-world problem, based on my sock experience:

If there are 10 socks in a load of laundry that I place in the washer and then transfer to the dryer, how many socks will remain when the load is finished drying?

**Nine.** Yes, I *know* it’s a cliche. Everyone makes jokes about losing socks. It’s so overdone that the National Comedian’s Guild has declared a moratorium against them. But, *I’m not joking*. I can’t remember the last time I did a load of laundry and wasn’t missing a sock. I now have a drawer filled with unmatched socks, each like Tiger Woods longing for the return of its Lindsey Vonn.

Sadly, this post is going public just a little too late. **Lost Socks Memorial Day** was **May 9**, so we just missed that one. Likewise, we missed **No Socks Day** on **May 8**. But there are other holidays in the coming months when you can celebrate the amazing undergarments that protect our feet from our shoes:

- July (exact date TBD):
**Red Socks Day**(commemorating Sir Peter Blake) - October 4:
**Odd Socks Day**(Australia) - January (every Friday):
**Snow Sock Day**

And though not an official holiday, there are unlimited **Crazy Sock Days** happening at elementary, middle, and high schools near you.

99% of socks are single, and you don’t see them crying about it.

How do engineers make a bold fashion statement?

They wear their dark grey socks instead of the light grey ones.Somewhere, all of my socks, Tupperware lids, and ball point pens are hanging out together, just laughing at me.

—

Because I know you won’t be able to sleep tonight…

- I need to remove
**8 socks**from my sock drawer to guarantee a color match. - I don’t actually own socks in
**every color**of the rainbow. Just most colors. - The probability of selecting two socks from my drawer and getting a matching pair is
**23/140**.

### Colored Pyramids and the Mind of a 7-Year-Old

This is what kept me up last night. Literally.

Form a row of ten squares, with each square randomly colored red, green, or yellow. Call this Row 1. Then place nine squares in Row 2 slightly offset above Row 1, and color the squares in Row 2 according to the following rules:

- If two side-by-side squares have the same color, the square between them in the row above has the same color.
- If two side-by-side squares have different colors, the square between them in the row above has the third color.

Continue in this manner with eight squares in Row 3, seven squares in Row 4, and so on, with just one square in Row 10.

Here’s an example of a pyramid constructed in this manner:

The 1’s, 2’s and 3’s in the diagram appear because this image was created with Microsoft Excel. Formulas were used to determine the number in each square, and conditional formatting was used to color the squares.

So far, this is just a test of how well you can follow rules, and it isn’t much fun. But here’s where it gets interesting.

**How can you predict the color of the lone square in Row 10 after seeing only the arrangement of squares in Row 1 (and without constructing the rows in between)?**

That’s right…

*This shit just got real.*

I found this problem last night on the Purdue Math Department’s Problem of the Week website. I lay awake in bed longer than I should have, but no solution came to me either while laying there awake or while I was sleeping. When I woke up, I shared it with my sons. I suspected they’d have fun coloring squares; I never suspected what actually happened.

I explained the problem to Alex and Eli, and I showed them an example of how to generate Row 9 from an arbitrary Row 10. They then pulled out their box of crayons and constructed rows 8, 7, 6, …, 1. Alex then looked at his pyramid for about 8 seconds and said, “Oh, I get it. You can find the color of the top square by _________.” (*Spoiler omitted.*)

“Is that your conjecture?” I asked.

“What’s a *conjecture*?” he replied.

“It’s a guess,” I told him. “It’s what you think the rule is.”

“No,” he said. “It’s *not* a guess. That’s the rule.”

He was pretty cocky for a seven-year-old.

So we tested his rule for another randomly-generated Row 10. It worked. So we tested it again, and it worked again. We tested it for six different hand-drawn pyramids… and it worked for every one of them.

That’s when I generated **this Excel file**. We used it to test Alex’s conjecture on 100+ other pyramids. It worked every time.

I still have no idea how he divined the rule so quickly.

For me, though, the cool part came when I was able to extend the puzzle with the following:

**For what values of k can you predict the color of the lone square in Row k when there are k blocks in Row 1?**

Trivially, if I gave you just two blocks in Row 1, you could most certainly predict the color of the square in Row 2.

But if I gave you, say, five blocks in Row 1, could you predict the color of the lone square in Row 5?

The final part of the Puzzle of the Week description from the Purdue website says exactly what you’d expect it to say:

Prove your answer.

I have a proof showing that Alex’s conjecture holds. Incidentally, that proof can be extended to prove the general result for the extension just posed.

Alex, however, has not yet generated a proof of his conjecture.

Then again, he’s only seven.