Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 MMATHS Individual Round Problems/Problem 4"

(→‎Solution 1)
m (→‎Solution 1)
 
Line 8: Line 8:
 
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
 
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
  
−
==Solution 1==
+
==Solution==
 
It would be helpful to list some two-digit perfect squares. These are <math>16, 25, 36, 49, 64,</math> and <math>81</math>. We can eliminate <math>16</math> and <math>64</math>. Let's check each of the next ones.  
 
It would be helpful to list some two-digit perfect squares. These are <math>16, 25, 36, 49, 64,</math> and <math>81</math>. We can eliminate <math>16</math> and <math>64</math>. Let's check each of the next ones.  
  

Latest revision as of 23:44, 24 August 2023

Problem

Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit perfect square!"

Claire asks, "If I picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there a chance I'd know for certain what it is?"

Cat says, "Yes!" Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn't know my favorite number!

Claire says, "Now I know your favorite number!" What is Cat's favorite number?

Solution

It would be helpful to list some two-digit perfect squares. These are $16, 25, 36, 49, 64,$ and $81$. We can eliminate $16$ and $64$. Let's check each of the next ones.

For $25$, we have $5-2 = 3$, $5+2 = 7$.

For $36$, we have $6-3 = 3, 6+3 = 9$.

For $49$, we have $9-4 = 5, 9+4 = 13$.

For $64$, we have $6-4 = 2, 6+4 = 10$.

We can clearly see that $25$ doesn't work because $7$ would identify it, nor does $49$ because $13$ would identify it, nor does $64$ because $10$ would identify it. We're left with only $36$. Therefore, our answer is $\boxed {36}$.

-Arcticturn

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_MMATHS_Individual_Round_Problems/Problem_4&oldid=197406"