# Google’s doodle on Leonhard Euler’s birthday is ambiguously wrong

Today’s Google doodle is in celebration of Leonhard Euler‘s 306th birthday and I’ve reproduced it below:

Today’s question is:

What is wrong with this picture? Answer after the break…

The answer is that Google made a mistake with Leonhard Euler’s formula. There is more than one formula depending on whether the network (or what mathematicians call a graph) is on a plane or a three dimensional convex object.

The one for a network on a plane (which describes the inscribed triangle inside the first ‘O’ of Google as well as the doodle of a network below it and the representation of the “Bridges of Koenigsberg‘ problem to the left bottom) is

V – E + F = 1

where V is the number of vertices, E the number of nodes and F the number of regions

But Euler’s formula for networks on a sphere (such as the second ‘O’) or convex objects (such as the icosahedron inside the ‘G’) is

V – E  + F = 2

Naughty Google doodle. Could confuse people.

Of course contacting Google to point out a mistake is close to impossible, so that’s that.