Problem of the Week #51: Monday June 24th, 2024
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by this twitter post.
Consider the set of natural numbers 1 to 8. For this set of numbers, 6 is called the balance because the numbers before 6 add up to the same sum as the numbers after 6. In other words, 1+2+3+4+5 = 7+8.
a). For the set of natural numbers 1 to n, how do we find the balance number if there is one? b). For which natural numbers n does the set 1 to n have a balance number?
c). For which natural numbers n can the set 1 to n be divided into two sets of consecutive numbers that have the same sum?
d). For which natural numbers m and n can the set m to n be divided into two sets of consecutive numbers that have the same sum?
e). For which natural numbers n can the sum 1+2+…+n be written as the sum of another set of consecutive natural numbers?
f). Extending to geometry, in the right triangle below, where should we draw a line perpendicular to the base that divides the area into two equal parts?
g). In the right trapezoid below, where should we draw a line perpendicular to the bottom side that divides the area into two equal parts?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.