How many lines can you fit?
Place one short line in each triangle you use. Join two side-midpoints. Keep every line’s endpoints separate.
The clock starts when you reveal the puzzle.SIXTEEN TRIANGLES. ROOM FOR A SURPRISE.
Draw the lines.
Keep their ends apart.
How many fit?
Find the limit.
CHOOSE A TRIANGLE · THEN A LINE
Choose any triangle.
Line endpoint Shared endpoint—turn one of its lines
Choose a triangle, then tap a direction. Tap that direction again to erase it.AT YOUR OWN PACE
A brain snack, with room to think.
THE ANSWER OPENS ANOTHER DOOR
Now for the lovely part.
You have an answer. Explore how it works, then take the idea somewhere new.
See why it works ↓Keep your time
THE IDEA BEHIND YOUR ANSWER
Count the places
where lines can end.
Your arrangement gives a construction. Counting the available endpoints tells us why it cannot be beaten.
30 endpoints ÷ 2 = 15 lines
Every line needs two different endpoints. Once all thirty places are used, another line has nowhere to end.
A CONSTRUCTION AND A LIMIT
Three directions. Ten edges in each.
Look at the unit edges in one direction. Their rows contain 4, 3, 2 and 1 edges: ten in all. The other two directions contribute ten each. Each of these thirty edges has one midpoint.
Lines cannot share those midpoints. That gives an upper bound of fifteen lines. Your fifteen-line arrangement reaches the bound, so fifteen is the maximum.
Trying to mark all sixteen small triangles would require thirty-two separate endpoints. There are only thirty. The obstacle is the supply of endpoints, wherever the lines are drawn.
The idea to take away
Pair a construction with a bound. Showing what can be done and why more is impossible turns a good attempt into an optimum.