Student Solution to Test Yourself (Parity)

This post originally appeared on the Brilliant blog on 8/31/2012.

Students that are interested in submitting their solution to Test Yourself are welcome to do so, especially for questions marked with a (*). Submissions should be sent to challenges@brilliantscholars.com. If we present your solution, you will be acknowledged.

The following is a Student Solution to a Test Yourself posted on Parity, presented by Farrell W., Philippines.

Test Yourself 5: (*) In the forest, there are chameleons of 3 colors: blue, red and green. When two chameleons of a different color meet, they will change into the third color. For example, if a red and blue chameleon meet each other, they will both turn green. If there are currently 12 blue chameleons, 34 red chameleons and 56 green chameleons, would it be possible for all the chameleons in the forest to turn green?

Key Techniques: Invariance; Modulo Arithmetic

Solution: Let (Bt,Rt,Gt) (B_t, R_t, G_t) represent the number of blue, green and red chameleons in the forest at time t t, respectively. Let us look at the value GtRt G_t - R_t taken modulo 3. When a blue and red chameleon meet, the new value is Gt+1Rt+1=(Gt+2)(Rt1)=GtRt+3GtRt(mod3) G_{t+1}-R_{t+1} = (G_t+2)-(R_t-1) = G_t - R_t + 3 \equiv G_t - R_t \pmod{3} . When a red and green chameleon meet, the new value is Gt+1Rt+1=(Gt1)(Rt1)=GtRtGtRt(mod3) G_{t+1}-R_{t+1} = (G_t -1 ) - (R_t - 1) = G_t - R_t \equiv G_t - R_t \pmod{3} . When a green and blue chameleon meet, the new value is Gt+1Rt+1=(Gt1)(Rt+2)=GtRt3GtRt(mod3) G_{t+1}-R_{t+1} = (G_t -1) - (R_t +2 ) = G_t - R_t -3 \equiv G_t - R_t \pmod{3} . Hence, this value stays constant throughout, regardless of which chameleons meet each other. We say that the value GtRt(mod3) G_{t}-R_{t} \pmod{3} is an invariant of the problem.

At the start, GtRt=5634=221(mod3) G_t - R_t = 56 - 34 = 22 \equiv 1 \pmod{3}. If all chameleons are green, then GtRt=(56+34+12)00(mod3) G_t - R_t = (56+34+12) - 0 \equiv 0 \pmod{3}. Since these two values are not equal, it is not possible for all the chameleons to turn green.

Test Yourself

  1. (*) Can you describe all possible sets of (B,R,G) (B, R, G) such that we can have B B blue, R R red and G G green chameleons in this forest?

Note by Calvin Lin
8 years ago

2 votes

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