Number of moves to switch all tiles from black to red? 
Four tiles are arranged as per the diagram and all start off black. On each move, two connected tiles may be interchanged, and upon doing so each of the two tiles switches color from red to black or black to red depending on their current state. [As an example, swapping $a$ and $b$ would make them both red, and then swapping $a$ and $d$ would make $d$ red but make $a$ black again]. What is the minimum number of moves needed to change all the currently black tiles to red while still having them end up in the same position as they started?


After playing around with it, I found the answer was 6. However, is there a more mathematical way to approach this? The only insights I could think of was that each tile must move an odd number of times for it to swap color.

Hint: Each tile must be moved an odd number of times.
Hint: Each tile must be moved more than once.
Hence, each tile must be moved at least 3 times.
Hence, the number of moves is at least $ \frac{4 \times 3 } { 2} = 6 $.
We still need to establish that 6 is possible. Trial and error suffices.
