Geometry and Symmetry
Shadow Seminar #1
Boston University, Summer 2006
Exploration: Triangle Sums
P1: Use the numbers 1 through 6. Record a different number in each circle. The row sums must
be the same. How many different sums are possible? What is the least possible sum? What
is the greatest possible sum? Are all the whole number sums between the least and the
greatest possible? How many “different” solutions are there for a particular sum? How is
this problem related to one you have done before? What if the sums had to be the
consecutive integers 11, 12, and 13? … consecutive integers 8, 9, 10? … consecutive even
integers 8, 10, 12? … consecutive odd integers 9, 11, 13? …the triplet 7, 10, 13? …the
triplet 8, 11, 14? How many different triplets are possible? How many “different” solutions
are there for a particular triplet? How could you change the difficulty level of this problem?
P2: Change the diagram above to 4 circles on each side of the triangle and use the numbers 1
through 9? Record a different number in each circle. The row sums must be the same. How
many different sums are possible? What is the least possible sum? What is the greatest
possible sum? Are all the whole number sums between the least and the greatest possible?
How many “different” solutions are there for a particular sum? Can you still find
consecutive integer sums like in P2? How does this change the difficulty level of the
problem?
Counting Exploration
P3: The Annual Mathematics Contest presented an interesting puzzle last year. The rules of the
contest and overlapping memberships caused some complications.
Here are the facts.
• Each team in the contest was represented by four students.
• Each student was simultaneously the representative of two different teams.
• Every possible pair of teams had exactly one member in common.
How many teams were present at the contest? How many students were there altogether?
Here is a different set of facts for the Annual Mathematics contest.
• Each team in the contest was represented by four students.
• Each pair of students belongs to exactly one team.
• Every possible pair of teams had exactly one member in common.
How many teams were present at the contest? How many students were there altogether?
How could you change the difficulty level of this problem?
Here is a different set of facts for the Annual Mathematics contest.
• Each team in the contest was represented by three students.
• Each pair of students belongs to exactly one team.
• Every possible pair of teams had exactly one member in common.
How many teams were present at the contest? How many students were there altogether?
How could you change the difficulty level of this problem?
SED ME 580
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Carol Findell