An Alphabet contains only A and B. Can $ABB = BAA$?

by Angelo Mark   Last Updated August 14, 2019 06:20 AM

An Alphabet containing only $A$ and $B$.

Removing $AB$ makes no difference to a word and adding $BA$ or $AABB$ makes no difference to a word.

Is $ABB = BAA $?

Tags : contest-math


Answers 3


ABB can only be equal to words that have one more B than A. So BAA is not achievable.

Matthew Daly
Matthew Daly
August 14, 2019 05:40 AM

Observe that if two words have no difference, then the difference of the number of letter $A$ in the word and the number of letter $B$ must be same. In the problem, $ABB$ has the difference of -1 but $BAA$ is 1. Therefore, $ABB$ is not equal to $BAA$.

Isaac YIU Math Studio
Isaac YIU Math Studio
August 14, 2019 06:07 AM

Am I Correct?

$ABA=A$ (By removing $AB$)

Now by adding $BA$ will not make any difference, so

BA$ABA=A$

$BA$AB$A=A$

By removing $AB$ we get $BAA=A $(1)

Clearly , $ABB=B$ (2)(By removing $AB$)

Now by (1) and (2), we have $BAA=A$ and $ABB=B$.

Since $A$ and $B$ are unique, $BAA \neq ABB$

Angelo Mark
Angelo Mark
August 14, 2019 06:15 AM

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