How-To: Guess the number of M&Ms in a jar


Here’s how to guess the number of M&Ms in a jar using its “packing fraction” – it gets exciting around the 3:30 mark…

13 thoughts on “How-To: Guess the number of M&Ms in a jar

  1. Simon Kokosza says:

    Hello, I just watched this video and it was pretty interesting. I would like to add some questions to this. Does this take into account random close pack? If so how would you know if this was not randomly close packed? What if while this jar was being filled and shaken while filling? That would produce some different results.

    1. Anon says:

      I believe that’s discussed in the paper (but I don’t have access as I’m home for the summer). Basically you just end up with a different packing fractions (that can all be measured experimentally).

    2. Bradley Monakhos says:

      @Simon Kokosza

      This is actually brought up in the article mentioned in Phillip Torrone’s video above, “Improving the Density of Jammed Disordered Packings Using Ellipsoids” (see reference 1)

      Many experimental and computational algorithms
      produce a relatively robust packing fraction
      (relative density) = 0.64 for randomly
      packed monodisperse spheres as they proceed to
      their limiting density (8). This number, widely
      designated as the random close packing (RCP)
      density, is not universal but generally depends
      on the packing protocol (9). RCP is an ill-defined
      concept because higher packing fractions
      are obtained as the system becomes ordered, and
      a definition for randomness has been lacking. A
      more recent concept is that of the maximally
      random jammed (MRJ) state, corresponding to
      the least ordered among all jammed packings
      (9). For a variety of order metrics, it appears that
      the MRJ state has a density of 0.637 and is
      consistent with what has traditionally been
      thought of as RCP (10). Henceforth, we refer to
      this random form of packing as the MRJ state.

      We report on the density of the MRJ state
      of ellipsoid packings as asphericity is introduced.
      For both oblate and prolate spheroids,
       and Z (the average number of touching
      neighbors per particle) increase rapidly, in a
      cusp-like manner, as the particles deviate
      from perfect spheres. Both reach high densities
      such as 0.71, and general ellipsoids
      pack randomly to a remarkable 0.735,
      approaching the density of the crystal with
      the highest possible density for spheres
      (11) = pi/rad.18 = 0.7405. The rapid increases
      are unrelated to any observable increase
      in order in these systems that develop
      neither crystalline (periodic) nor liquid crystalline
      (nematic or orientational) order.

      Ref. 1
      Donev, A., Cisse, I., Sachs, D., Variano, E.A., Stillinger, F.H., Connelly, R., Torquato, S., Chaikin. P.M. (2004). Improving the Density of Jammed Disordered Packings Using Ellipsoids. Science, Vol. 303. no. 5660, pp. 990 – 993

    3. Charles says:

      I think you can apply the law of large number in this case and take the midpoint of the packing fraction range and use that as the value to calculate the number of m&m’s in the jar. After all it’s an estimate.

  2. David Baker says:

    I watched this video and understand what he is saying, but hoped to end up with an equation that would help me win the M&Ms. Can someone put together that equation for me?

    1. smokeythebear says:

      In order to calculate how much M&M’s would be needed to fill up the entire jar, you would take the “volume of the jar”/”volume of one m&m”

      This value would give you how many m&m’s it would take to fill the jar completely, that’s excluding the spaces of air in between the m&m’s. This is where the packing fraction comes in. It pretty much says how much of the total volume of the jar is taken up my only the m&m’s. In this case, from the paper, it was 66.5%. This means that 33.5% is made up of air in between the m&m’s.

      So your final equation to figure out the number of m&m’s is:


      Remember that the packing fraction is different types of candies. IE, peanut mm’s will have a different packing fraction from regular ones.

  3. Otto says:

    Yeah, that works perfectly… Until somebody opens the thing up and finds a partially inflated balloon in the middle of the M&Ms.

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