### Video Transcript

Three resistors, each with a resistance π
, are connected in different ways as shown in the diagrams one through four. Which figure gives the smallest value of resistance between π and π?

In figures one, two, three, and four, we see these resistors arranged between points marked π and π in each diagram. To solve for the figure which gives the smallest value of resistance between these two points, we can redraw each one. Figure one can be redrawn to show π then a single resistor followed by two resistors arranged in parallel then point π.

Figure two can be redrawn as π and π separated by two parallel branches one with resistance π
and the other with resistance two π
. In figure three, π and π are separated by a resistance of π
. And there is a resistance of two π
that leads to a break in the circuit. And finally, in figure four, we can combine all three of the resistors in series so that π and π are separated by three π
of resistance.

Now that weβve drawn simplified circuits for each diagram, we can calculate the resistance between points π and π in each one. In diagram one, π
ππ is equal to π
, the value of the resistor in series, plus the value of the combined parallel resistors which is π
squared, their product, divided by two π
, their sum. In this fraction, one factor of π
cancels out so that π
sub ππ is equal to π
plus π
over two or three-halves π
. Thatβs for diagram one.

For diagram two, π
sub ππ is equal to the product of the two resistances in parallel: two π
times π
all divided by their sum which is three π
. Once again, we see a factor of π
cancel from numerator and denominator. In total then, π
sub ππ for diagram two is two-thirds π
.

Moving on to diagram three, in this case, π and π are separated by one resistor of resistance value π
. We write down that value then for π
sub ππ for this diagram and move on to our final diagram number four. In this final diagram, π
sub ππ is equal to three π
.

Comparing these four values, we find that π
sub ππ equalling two-thirds π
is the smallest of the bunch. That means the figure with the smallest resistance between points π and π is figure two.