Problem 26 · 2022 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintcasework
The numbers 1 to 10 were written into the ten circles in the pattern shown in the picture. The sum of the four numbers in the left and the right column is 24 each and the sum of the three numbers in the bottom row is 25. Which number is in the circle with the question mark?
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Answer: E — another number
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Hint 1 of 2
All ten numbers add to 55; the two columns already use 24 + 24 = 48.
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Hint 2 of 2
That leaves 7 for the two circles outside the columns; the bottom-row total of 25 then pins each one.
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Approach: use total 55 and the column/row sums
- 1+2+···+10 = 55. The left and right columns (four each) take 24 + 24 = 48.
- The remaining two circles sum to 55 − 48 = 7; with the bottom row equal to 25 they are forced to be 6 and 1.
- The question-mark circle is the one equal to 1, which is none of 2, 4, 5, 6, so the answer is another number.
Mark:
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