Alice in Wonderland -- height riddle

Algebra Level 3

The height of Alice a a (in meters) satisfies the equation 2 a + 2 a = 5 2a+2^a=5 .

The height of Mad Hatter m m (in meters) satisfies the equation 2 m + 2 l o g 2 ( m − 1 ) = 5 2m+2log_{2}(m-1)=5 .

You may not know their exact height, but we can derive some relationships.

What is the value of a + m a+m ?

Of course, in wonderland we don't have a calculator, nor Wolfram|Alpha.


The answer is 3.5.

Alice Smith by Alice Smith , 20, China

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1 solution

Let m=1+ 2 n 2^n . Then the two given equations give n=a-1. Hence a+m=a+1+ 2 n 2^n =1+ 1 2 \dfrac{1}{2} (2a+ 2 a 2^a )=1+ 5 2 \dfrac{5}{2} =3.5

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