About Exercise 6.8:

Make a list line plot of the number of digits in each of the first 100 squares.

One of the possible solutions:

ListLinePlot[Table[Length[IntegerDigits[i^2]], {i, 1, 100}]]

Another solution:

ListLinePlot[
 Table[Length[IntegerDigits[Range[100]^2][[i]]], {i, 100}]]

Hint (and only a hint):

You have a list of items and you want to do the same thing to every one of those items.

Think "Map" and then think what is that thing that you want to do to every one of those individual items (which just happen to be lists).

About the notation @. It is so named prefix form, i.e. you may use notation f[x] or f@x. They are identical.

For example:

Plot[Sin[x], {x, -7, 7}]

is identical to:

Plot[Sin@x, {x, -7, 7}],

i.e. Sin[x] and **Sin@x** are identical.

About the notation /@. It stands for the function Map[].

For example, you can write:

In[1]:= Map[f, {a, b, c, d, e}]

Out[1]= {f[a], f[b], f[c], f[d], f[e]}

    or in alternative input form:

In[2]:= f /@ {a, b, c, d, e}

Out[2]= {f[a], f[b], f[c], f[d], f[e]}

P.S. The first solution without the symbol @ is equivalent to

ListLinePlot[
 Table[First[Part[IntegerDigits[Table[i^2, {i, 100}]], j]], {j, 100}]]

Thank you for in-depth answer above, however, I did not know syntax you used {embedded table formula and use of @ and in this example, this reply, /@. I found Arno Bosse's answer easier. (maybe because that was the solution I came upon first).

By the way, I am stuck at Question 8 of same chapter of the book. Any help?

Other variations on the same subject of Exercise 6.10:

ListLinePlot[IntegerDigits[Range[100]^2][[;; , 1]]]
IntegerDigits[Range[100]^2][[;; , 1]] // ListLinePlot    
ListLinePlot[First /@ IntegerDigits[Range[100]^2]]
First /@ IntegerDigits[Range[100]^2] // ListLinePlot
Table[IntegerDigits[i^2][[1]], {i, 100}] // ListLinePlot

That's interesting. I didn't understand at first how this could work, but

IntegerDigits[Range[100]^2] // TableForm

makes it clear. From all rows, select the first column.

Nice solution!

Thanks for the replies.

Apparently this is the solution, but the output is clearly not correct(according to the solution), so I need to make it work.

, 100}]]rst /

This is how I imagine the solution would be like, using functions that were taught in the 5 previous chapters of "An Elementary Introduction to the Wolfram Language".

Edit: I've already spent 2 days on this question, so it's time to move on, otherwise I will take a year or so to complete this online book. Maybe Stephen Wolfram(the author) expects too much of the reader, if he thinks that they can answer that question using only what has been taught previously. :)