Answer to the Problem of the Week
for the week of March 13, 2006

Fourteen Questions
idea by Michael Sontchi
in Games Magazine (August 2000)

Each of the 14 questions below is answered by a different whole number from 1 to 14.   Can you figure out which number goes with each question?

  1. How many of these questions have an answer that isn't a perfect square?
  2. How many of these questions, if their answers were considered atomic numbers, would represent elements that end in the letter M?
     
  3. How many of these questions have an answer that couldn't be the length of one side of a right triangle, if each side's length is an integer from 1 to 15?
  4. How many of these questions have an answer that could be the third side of a triangle if the other two sides' lengths are 8 and 10?
     
  5. How many of these questions have an answer that divides evenly into 27,720?
  6. How many of these questions have an answer that divides evenly into another answer?
     
  7. How many of these questions have an answer that divides evenly into the sum of all the answers?
  8. How many of these questions have an answer that, if converted to a binary number, is not a palindrome?
     
  9. How many of these questions have an answer that is a rational number?
  10. How many of these questions have an answer that is a perfect cube?
     
  11. How many of these questions have an answer that is a perfect number?
  12. How many of these questions have an answer that is both an odd prime and less than the average of all the answers?
     
  13. How many of these questions have an answer that is divisible by 2 or 7?
  14. How many of these questions have an answer whose factorial ends in 0?
     


 

Solution to the Problem:

a     11     All except 1, 4, and 9
b     6     2 (Helium), 3 (Lithium), 4 (Beryllium),
  11 (Sodium), 12 (Magnesium), and 13 (Aluminum)
 
c     5     1, 2, 7, 11, and 14
d     12     All except 1 and 2
 
e     13     All except 13
f     7     1, 2, 3, 4, 5, 6, and 7
 
g     4     1, 3, 5, and 7
h     9     2, 4, 6, 8, 10, 11, 12, 13, and 14
 
i     14    
j     2     1 and 8
 
k     1     6
l     3     3, 5, and 7
 
m     8     2, 4, 6, 7, 8, 10, 12, and 14
n     10     5, 6, 7, 8, 9, 10, 11, 12, 13, and 14

Thanks to Jim Arrison and Magdy Essafty for pointing out my error in the wording of clue M (it originally asked for the numbers divisible by BOTH 2 AND 7).

Also, thanks to Larry Schwartz for pointing out the error for clue C.
Originally, Question c. stated "How many of these questions have an answer that couldn't be the length of one side of a right triangle, if each side's length is an integer?" Larry wrote that "All integers from 3 to 14 could be a side of a right triangle:
3 4 5
6 8 10
7 24 25
9 12 15
11 60 61
5 12 13
14 48 50

I changed the wording so that the sides of the right triangle had to be integers from 1 to 15, so that eliminated 3 of the triangles above.

My apologies to those who printed the original problem and then struggled with making it work.


Correctly solved by:

1. Sagar Patel Brookstone School
Columbus, Georgia
2. Jeff McKenzie John Handley High School
Winchester, Virginia
3. Andrew Dau South O'Brien High School
Gaza, Iowa
4. Magdy Essafty Alexandria, Egypt
5. Mike Singer John Handley High School
Winchester, Virginia
6. Larry Schwartz Norwalk, Connecticut
7. Sharina Smith Newport News, Virginia
8. Tristan Collins Virginia Tech
Blacksburg, Virginia
9. Billy Sutherland John Handley High School
Winchester, Virginia
10. Garrett Hix Brookstone School
Columbus, Georgia
11. Danny Gordon Brookstone School
Columbus, Georgia
12. Julianne Harris John Handley High School
Winchester, Virginia
13. Maggie Morrison John Handley High School
Winchester, Virginia
14. David & Judy Dixon Bennettsville, South Carolina
15. Andrew Montoya John Handley High School
Winchester, Virginia
16. Ben Bassett Sewell, New Jersey
17. Kelley Kolar John Handley High School
Winchester, Virginia
18. Emily Stapp Mountain View High School
Mountain View, Wyoming