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Because there are a large number of ways in which transversals can be
constructed in the T-hexagon of side 3, we asked the Year 8-9 Groups to
explore what transversals they could find with position 1 in Row 1
occupied. This is one of the team solutions:
SEE: In this question, we should position another 5 counters in the
hexagon.
PLAN: We marked all of the triangles at first with letters for rows
and numbers for positions. Then we had a trial run.
DO: The question has positioned a counter in A1. We only can put the
counter in Row B in the triangles B4-B9 because we can't put 2 counters
in one horizontal or diagonal row. We begin from A1, B4. We made a
bigger picture and used coins to show the positions of the counters.
First, the counter in A1 means that the other counters can not be
positioned in the red area of Row B, So the counter in Row B only can
put in the green area. If we put the counter in B4, then the red area
will fill out (second picture). Then we'll try the counters in Rows C,
D, E and F.
CHECK: At last, we found the first answer. We then found many more answers
by trying the same method. Here are the number of solutions we found for
each of the Row B positions:
| Positions in A and B |
# Solutions |
| A1, B4 |
10 |
| A1, B5 |
4 |
| A1, B6 |
10 |
| A1, B7 |
7 |
| A1, B8 |
10 |
| A1, B9 |
9 |
| Total |
50 |
|
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Note: The problem as set (because of the time constraint) only asked the
students to examine what happens if the A1 position is occupied. If
position A2 is occupied, the configuration can be reflected in the
top-left to bottom right diagonal, and thus there will be another 50
possibilities. Similarly, if positions A6 and A7 are occupied,
reflections and rotations can always bring that cell to the A1 position.
Thus there are 200 transversals that use cells A1, A2, A6 and A7 of Row
1. If A3 is occupied, it turns out that there are 18 possibilities,
which will be the same for A5. Finally, with position A4 occupied, there
are another 28 transversals. This makes (we think) 264 transversals in
all.
However, the process of eliminating transversals that are the same as
others after rotation or reflecting reduces the options. We
have found only 35 transversals that are unique or non-congruent.
Here we borrow the geometric definition of congruency as meaning that
two figures are congruent if one can be rotated, translated and/or
reflected into the other. |
