Inspired by Maggie Miller

Probability Level 5

You have a large jar filled with colored marbles. It contains 2013 blue 2013 \,\color{#3D99F6}{\text{blue}} marbles, 2014 red 2014\,\color{#D61F06}{\text{red}} marbles, 2015 orange 2015\,\color{#EC7300}{\text{orange}} marbles, 2013 yellow 2013 \,\color{#cccc00}{\text{yellow}} marbles, 2014 purple 2014\,\color{#69047E}{\text{purple}} marbles, and 2015 green 2015\,\color{#20A900}{\text{green}} marbles.

You draw four marbles out of the jar, then put three marbles in (decreasing the total number of marbles in the jar by one) using the following rule:

  • If the four marbles are 1 blue 1\,\color{#3D99F6}{\text{blue}} marble, 2 red 2\,\color{#D61F06}{\text{red}} marbles and 1 orange 1\,\color{#EC7300}{\text{orange}} marble, put 2 yellow 2\,\color{#cccc00}{\text{yellow}} marbles and 1 green 1\,\color{#20A900}{\text{green}} marble into the jar.

  • If the four marbles are 2 orange 2\,\color{#EC7300}{\text{orange}} marbles and 2 purple 2\,\color{#69047E}{\text{purple}} marbles, put 1 red 1\,\color{#D61F06}{\text{red}} marble, 1 yellow 1\,\color{#cccc00}{\text{yellow}} marble and 1 green 1\,\color{#20A900}{\text{green}} marble into the jar.

  • If the four marbles are 3 green 3\,\color{#20A900}{\text{green}} marbles and 1 red 1\,\color{#D61F06}{\text{red}} marble, put 1 blue 1\,\color{#3D99F6}{\text{blue}} marble, 1 yellow 1\,\color{#cccc00}{\text{yellow}} marble and 1 purple 1\,\color{#69047E}{\text{purple}} marble into the jar.

  • If the four marbles are 2 yellow 2\,\color{#cccc00}{\text{yellow}} marbles and 2 green 2\,\color{#20A900}{\text{green}} marbles, put 1 red 1\,\color{#D61F06}{\text{red}} marble, 1 orange 1\,\color{#EC7300}{\text{orange}} marble and 1 purple 1\,\color{#69047E}{\text{purple}} marble into the jar.

  • If the four marbles are 3 purple 3\,\color{#69047E}{\text{purple}} marbles and 1 yellow 1\,\color{#cccc00}{\text{yellow}} marble, put 1 blue 1\,\color{#3D99F6}{\text{blue}} marble and 2 green 2\,\color{#20A900}{\text{green}} marbles into the jar.

  • If the four marbles are 4 green 4\,\color{#20A900}{\text{green}} marbles, put 2 orange 2\,\color{#EC7300}{\text{orange}} marbles and 1 blue 1\,\color{#3D99F6}{\text{blue}} marble into the jar.

  • In all other scenarios, return all four marbles into the jar.

Repeat this process until one marble remains. What color is it?

Click here for the inspiration problem.
Yellow Purple Red Orange Green Blue

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Khang Nguyen Thanh by Khang Nguyen Thanh , 27, Vietnam

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1 solution

Let ω = 1 + i 3 2 \omega=\dfrac{1+i\sqrt{3}}{2} .

We denote a blue \color{#3D99F6}{\text{blue}} marble by number 1 1 , a red \color{#D61F06}{\text{red}} marble by number − 1 -1 , a orange \color{#EC7300}{\text{orange}} marble by number ω \omega , a yellow \color{#cccc00}{\text{yellow}} marble by number − ω -\omega , a purple \color{#69047E}{\text{purple}} marble by number ω 2 \omega^2 , and a green \color{#20A900}{\text{green}} marble by number − ω 2 -\omega^2 .

Let T k T_k is product of all numbers denoted in the jar when k k marble(s) have been removed.

Note that T 0 = ω 2 T_0=\omega^2 and T k + 1 = T k T_{k+1}=T_k for all integer k k .

This implies that T 12083 = T 0 = ω 2 T_{12083}=T_0=\omega^2 .

So, the color of remained marble is Purple \boxed{\text{Purple}} .

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