Problem 29 · 2012 Math Kangaroo
Stretch
Arithmetic & Operations
arithmetic-seriescareful-counting
The natural numbers from 1 to 120 were written as shown into a table with 15 columns. In which column (counting from left) is the sum of the numbers the largest?
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Answer: B — 5
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Hint 1 of 3
The picture is a triangle: row 1 has one number, row 2 has two, … row 15 has fifteen (since 1 + 2 + … + 15 = 120).
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Hint 2 of 3
Column j is only filled from row j downward, so far-left columns have many small numbers and far-right columns have few large ones — the winner is somewhere in between.
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Hint 3 of 3
Compare a column's total to its right neighbour: moving right drops one small top entry but adds 1 to every entry below it.
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Approach: see the triangular fill, then balance 'fewer entries' against 'bigger entries'
- Row n holds n numbers, ending at 1 + 2 + … + n; row 15 ends at 120, so it is a 15-row triangle and column j is filled only in rows j through 15.
- Column 1 has fifteen entries but they are the smallest in each row; the far-right columns have only a few entries even though they are large — so the biggest total sits in the middle.
- Adding up each column gives totals 575, 588, 598, 604, 605, 600, 588, … which peak at column 5.
- So the largest column sum is in column 5 (B).
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