Problem 12 · 2011 Math Kangaroo
Hard
Algebra & Patterns
sum-constraint
Michael wants to write whole numbers into the empty cells of the 3×3 table on the right so that the numbers in every 2×2 square add up to 10. Four numbers are already filled in. Which of the following could be the sum of the remaining five numbers?
Show answer
Answer: E — None of these numbers is possible.
Show hints
Hint 1 of 3
Call the centre cell c and write each 2×2 sum-equals-10 condition in terms of c.
Still stuck? Show hint 2 →
Hint 2 of 3
Add the four border cells and the centre — the total comes out as 20 − 3c.
Still stuck? Show hint 3 →
Hint 3 of 3
Check whether any listed value has the form 20 − 3c for a whole number c.
Show solution
Approach: express the five unknowns through the centre value
- With the centre = c, the four 2×2 conditions give the corners as 7−c, 5−c, 5−c, 3−c.
- Their sum with c is (7−c)+(5−c)+c+(5−c)+(3−c) = 20 − 3c.
- That is always 2 more than a multiple of 3, but 9, 10, 12, 13 are not.
- So none of these values is possible.
Mark:
· log in to save