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
CAN YOU FIND THE VALUE OF X IN THE FOLLOWING
EXPRESSION?
$$\log_{2}{x}+\log_{2}{x} =5$$
Puzzled?
Need a refresher on logarithms? Watch this video.
Try applying the product property of Logarithms!
You are about to leave for holiday, but you forgot your socks!
You race back to your room, but are too lazy to flip on the lights… so you can’t see the color of the socks.
Fortunately, you remember that in your drawer, there are ten pairs of green socks, ten pairs of black socks, and eleven pairs of blue socks, but they are all mixed up.
How many of your socks do you need to take before you can be sure to have at least one matching pair?
Puzzled?
What matters is how many different colors there are, not how many of each color!
Try applying something called the Pigeonhole Principle. If you don’t know what that is, watch this video!
ANSWERS TO LAST WEEK’S PUZZLERS
Last Week's Puzzler Quik
Carter from San Diego, California was the winner of this shout-out.
We compute each place value of the given base 2 integer in base 10.
The right-most place value is 2^0=1, the second place value is 2^1=2, the third place value is 2^2=4, and the fourth is 2^3=8.
Thus, the base-10 equivalent of the given base-2 integer is $$1\cdot1+2\cdot1+4\cdot0+8\cdot1=11.$$
Last Week's Puzzler Think
This week’s shout out goes to Cheryl and Collin from Wisconsin! Kudos to the duo for solving the problem!
We convert the two numbers to base-10 as follows, and set them equal:
$$c^3+c^2+c+1=6(c+1)^2+2(c+1)=>c^3−5c^2−13c−7=0.$$
Since c must be an integer, we use the Integer Root Theorem to test for roots to this cubic polynomial – the solutions are c=7 and c=-1, but since the latter is extraneous, the answer is 7.
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