Welcome to The Puzzler. Every week, there are 2 new puzzles related to my passion: math, logic, and thinking. The first puzzle will be The Puzzler Quik, meant for those who crave something fun-sized. The second puzzle will be The Puzzler Think, meant for those who love to ponder. The answers will be posted in next week’s column. Don’t forget to submit your answer for a potential shout-out in the next edition of the Puzzler.
Puzzler Quik
What is 10112 in decimal(base 10)?
Puzzled?
If you don’t know what binary numbers are, learn about them here!
No second hit this time!
If $$1111_{c} = 620_{c+1},$$ find the value of c!
Puzzled?
Try rewriting both sides with a common base!
The common base should be base 10.
ANSWERS TO LAST WEEK’S PUZZLERS
Last Week's Puzzler Quik
Ari from Colorado was the winner of this shout-out.
We just calculate each log separately!
$$\log_{23}{23} = 1$$ $$\log_{23}{1} =0$$ $$\log_{16}{256}=2$$
Thus, our answer is 1 + 0 +2 = 3.
If you don’t know what logarithms are, just watch the following video.
Last Week's Puzzler Think
This week’s shout out goes to Lalit from New Jersey! Kudos to him!
We use the change of base formula to show that![\[\log_a b = \dfrac{\log_b b}{\log_b a} = \dfrac{1}{\log_b a}.\]](https://latex.artofproblemsolving.com/f/7/3/f73e271cefc08bdc7f68146a8b825b7186a45f32.png)
Thus, our equation becomes![\[\log_a 2 + \log_a 3 + \log_a 4 = 1,\]](https://latex.artofproblemsolving.com/1/0/2/10290ea33d9b8f7e8c2fce22f914ab6872c10f8a.png)
which becomes after combining:![\[\log_a 24 = 1.\]](https://latex.artofproblemsolving.com/c/d/3/cd3467e85a1cde569caa3051ba6a69ec140d9b84.png)
Hence 
(2015 AMC 12A Problem 14)
Pingback: The Puzzler – Week 26 -