logical deduction - Create a Word Search

Create a word search with the following conditions:

1. Use the eleven words below. They can be spelled up, down, left, right, or any of the 4 diagonal directions.

ZERO
ONE
TWO
THREE
FOUR
FIVE
SIX
SEVEN
EIGHT
NINE

TEN

2. 
   

   The size of the word search must be 5 rows and 6 columns.

   
   


3. 
   

   The number of words that pass through each square must fit this pattern:

   
   

4. The accepted answer will clearly and fully explain the flow of logic used from start to finish.

Answer

Here's my answer:

T E N I N E
F I O N E V
O G X R V I
U H H I E F
R T W O S Z

I started by figuring out the letter density in the overall puzzle. The letters break down as follows:

E   10
N   5
I   4

O   4
T   4
R   3
F   2
H   2
S   2
V   2
G   1
U   1
W   1

X   1
Z   1

Next I started working on the 3 squares. Since all three are in a diagonal, I started with the assumption that they are within the same word. This requires a word of length 5, which means it is one of: SEVEN, EIGHT, THREE

Only one of those includes three of the letters that occur at least three times from above, so that line became THREE

Since H only occurs twice, I ran the THREE diagonally up and from the left.

.....E
....E.
...R..
..H...

.T....

First thought would be to place the EIGHT, as that shares the only other H, but that doesn't fit. H is on a single letter space, so let's leave that for now.

Looking at the 2 and 2 3s in the upper right corner, it seems like there should be quite a bit of overlapping there. Let's try placing the NINE and TEN as follows:

..NINE
....E.
...RT.
..H...
.T....

Not bad, could be promising.

So far, we have the following for letter counts

..1122
....2.
...11.
..1...
.1....

Uh-oh! We only have one end N left and it's on SEVEN. That won't fit in our open space. Let's try putting SEVEN where TEN is.

..NINE

....E.
...RV.
..H.E.
.T..S.

..1122
....2.
...11.
..1.1.
.1..1.

How about we run the EIGHT in the second column?

.ENINE
.I..E.
.G.RV.
.HH.E.
.T..S.

.11122
.1..2.

.1.11.
.11.1.
.2..1.

Ooh, I see almost all of TEN. Let's finish that.

TENINE
.I..E.
.G.RV.
.HH.E.
.T..S.


122122
.1..2.
.1.11.
.11.1.
.2..1.

Now, we can add TWO off the T from THREE and ONE off the first E

TENINE
.IONE.

.G.RV.
.HH.E.
.TWOS.

122122
.1113.
.1.11.
.11.1.
.3111.

Hey, let's finish off ZERO

TENINE
.IONE.
.G.RV.
.HH.E.
.TWOSZ

122122
.1213.
.1.21.

.11.2.
.31111

And we can run FIVE up the right side and FOUR down the left. Note, FOUR could be either up or down, whatever floats your boat.

TENINE
FIONEV
OG.RVI
UHH.EF
RTWOSZ


122123
112131
11.211
111.21
131111

And finish off with SIX

TENINE
FIONEV
OGXRVI

UHHIEF
RTWOSZ

122123
112131
111211
111121
131112