How many M&M’s are in the jar?

Last Thursday, there was a company-wide party. That morning, they put a big jar full of M&M’s (regular, not peanuts) at the lobby of the Truchard Design Center. You were supposed to estimate how many M&M’s were in there. Whoever makes the best guess, wins a $50 Gift Certificate at Best Buy (and everlasting glory).Continue reading for details on the complicated process involved in this chocolatey adventure.Step 1: What is the size of an M&M?

Find a caliber (Gustavo Castro let me use his digital caliber) and measure a bunch of chocolates. Average the sizes, calculate the volume of a single M&M. In case you’re curious: .636 cubic centimeters.

The measurements could not be repeated for verification… sample chocolates were eaten.

Step 2: What is the size of the jar?

There were no special rules about not getting near the jar. Take some measuring tape and get the perimeter of the jar, then the height of the volume occupied by M&M’s (not counting air between them – yet). Measure the thickness of the glass. Don’t forget concave base and the round corners of the jar. Then calculate the volume.

This is the jar full of m&m’s

Step 3: How much volume do the M&M’s really occupy?

Because of their unique shape, M&M’s occupy a percentage of the volume of the container when poured randomly. I didn’t calculate this… physicist Paul Chaikin and chemist Salvatore Torquato did all the hard work. I remember reading about it months ago, and Google helped me find their study. So, m&m’s poured randomly occupy 68% of the volume calculated above.

Paul Chaikin (left), professor of physics, and Salvatore Torquato, professor of chemistry, used M&M’s candies to reveal fundamental principles governing the random packing of particles at Princeton University. Photo: Denise Applewhite (2004).

With all these numbers, I came up with my answer…. 8608.9669. So I wrote 8608 and submitted my answer.

To the astonishment of everyone who saw me measuring chocolates that morning, I was almost dead on.

So how many M&M’s were really there?

The number of M&M’s in the jar was: 8609. I missed by one. I stupidly rounded down… unbelievable. Fortunately I still won the contest.

And for those who care… this is the breakdown by color (aproximate):

  • 1119: Brown
  • 1205: Yellow
  • 1119: Red
  • 2066: Blue
  • 1722: Orange
  • 1371: Green

[Update October 30th, 2007] Seriously people, I posted detailed instructions on how to calculate this thing. I’m not going to go solve your version of the problem if you request in the comments, no matter how nicely you ask. Now, if you send me a nice PayPal donation for the maintenance of this site, I just might :).

74 thoughts on “How many M&M’s are in the jar?”

  1. I am still trying to figure out how many peanut m&ms will fit in a one gallon glass jar. The glass jar is the type with the small ine inch spout witht he small hooped handle . The type apple cider or vinager usually comes in. Any help would be appreciated

  2. i don’t understand anything !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

  3. Ha, thank you so much. They are doing this at my school so I was googling the volume of an m&m and got this. Now I have to find one for skittles, jelly beans, and kissables.

  4. Haha, good job. If I were running such a competition, however, I would not let people take physical measurements of the jar: the whole point is to make them perform estimations by eye. Even so, I’m surprised you were so accurate, and I defend your rounding down, because you had to assume your model would work and there were no “half M&Ms” in the jar.

  5. Aha, you said you rounded down. You said that like it was a mistake. Actually, it wasn’t. If that 0.0331 of an m&m was extra and couldn’t fit inside the jar, you really don’t expect the people filling the jar to actually try to fit 1 more m&m in there… Because they aren’t THAT dedicated in putting that last m&m in there. I think it had something to do with your measurement, as that jar is pretty irregular!

  6. I’d be willing to pay a small “thinker” fee if you can get this calculated by 5pm est today 04-06-2010. Thanks

  7. What a great tutorial. If anyone is interested, there is also an iPhone app that does all the calculations for you for a number of candies and containers. I think it was “candy counter” or something like that. Worked for me.

  8. Thanks for all of the great info!

    For those of you a bit lost in the math… try this:

    1. Calculate the volume of your vessel in cubic centimeters. For a jar that is 20 cm across (radius=10cm) and 30 cm high, this would be:

    3.1416 x (10×10) x 30 = 9414.8 cubic cm

    2. Divivide the volume in cubic centimeters by 0.93529 (a factor based on the volume of the M&M, given in this article and the amount of dead space that each plain M&M consumes around it when randomly dumped in a jar). This should give you the approximate # of M&Ms in the jar!

    9414.8 cubic cm in the jar / 0.93529 = 10,066.18 M&Ms

    3. Guess 10,066 M&Ms

    4. Win M&Ms

    5. Eat entirely too much chocolate

    6. Come back to this thread and thank the author for assembling all of this information. Very cool. I enjoyed it very much!

  9. If the jar is labeled, can’t you just take the ml of the jar and divide by 0.93529?
    Example a 750ml bottle would contain 750 / .93529 = 802 m&ms

  10. Thank you! I used this to calculate out how many M&Ms to order to fill my favors for my wedding (which required the additional calculation of volume to weight since you order by weight).

  11. This made my day. Every year, my brother-in-law has some “guess how many thingies in here” contest at his Halloween party. Every year I come in second place….to my husband. Do you have any idea how irritating that is? Anyway, I wasn’t taking into account the volume occupied. Since I’m sure he doesn’t know that the ACTUAL percentage is 68% and has been winging that, I should be able to get him this year!

  12. Something about the cubic volume seems off. If you measure and average 13.3 mm and 6.8 mm and calculate volume using 4/3 pi a^2*b then you get 2.66 cm^2 not 0.636. Please clarify.

  13. I’ve got one… However it’s Mike and ikes container I believe is 10L (2 1/2 gallon). Other than that no details. 😔

  14. Hi Marcos! I was just wondering how many (regular) M&M’s could be in a 3 quart jar? I’ve tried to solve for an answer, but I’m not good at math. Can you help me? I would greatly appreciate it!

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