Guess which Izumi!

Solution:

Go on letter 'Z' to the left. From there, you can go either left, down or up, on a letter 'U'.

If left 'U' is chosen, there are 3 different ways to finish word if 'M' to the left is chosen, if down 'M' is chosen there are 2 ways, that is the same with upper 'M'. Meaning, 7 ways for that left 'U'.

Now, if upper 'U' is chosen there are 4 ways to finish it. It is completely similar for down 'U'. Both of them united, 8 ways.

Since all previously mentioned ways started with left 'Z' (15 ways), solution is obtained when multiplied with 4 (since it is the same if down, up or right 'Z' is chosen). The answer is $\boxed{60}$

We can use the classical "walking on a rectangular grid" problem to solve:

This is a quarter of the image. 'Cause the image is symmetric, we can calculate in a quarter and then multiply it by $4$ .

Starting from letter 'I', we need $4$ steps to reach the last 'I'. In each step, there are $2$ ways: up or right.

So the number of 'IZUMI's are $2^4=16$ . Multiply by $4$ gives us $64$ .

But, $4$ words 'IZUMI's are duplicate, which are, the vertical and horizontal ones. We need to subtract by $4$ .

The answer is $\boxed{60}$ .

vertical = v

horizontal = h

v or h : 1

vh : 3

hv : 3

vhv : 3

hvh : 3

hvhv : 1

vhvh : 1

sum = 15

total = 15*4 = 60