connect all the 9 dots on a 3 by 3 grid with just 4 straight lines without lifting your pencil (pen) off the paper (so a new line should be connected to the previous one)
how many ways can you find to complete this puzzle?
[discounting rotations and reflections there is one unique way to solve this problem - proved here]
(see also the 4 lines problem)
try to do the same thing with the 16 dots on a 4 by 4 grid
try to do this with just 6 straight (joined) lines
try to find more than one solution
can you do this and return to the start dot (i.e. with a closed journey)?
try to do the same thing on the 5 by 5 grid with 8 straight lines
how many lines for the 6 by 6 grid?
Danial Goodwin provides this connection method
both are symmetrical
an elegant method for connecting all dots on a 7 by 7 grid
if the grid is 1cm between dots, how long is this route (with root 2s in the answer)?
how many lines for a 2 by 3 grid?
how many lines to connect a 2 by 3 grid with a return to the start?
how many for a 3 by 4 grid?
try to find several different solutions, using 5 connected straight lines (including a simple one)
you can do the 3 by 4 problem with 5 lines and return to where you started (but this isn't easy)
and you can do the 4 by 5 grid and return to the start with 7 lines