How can I solve this logic question using propositional logic (Natural deduction)? $$\big((P\rightarrow Q)\rightarrow P\big) \rightarrow P$$
I need to solve this using simple natural deduction rules
these can be hypothesis, $\rightarrow$ intro, $\rightarrow$ elim, conj and disjunct intro and elim, and negation intro (Reductio ad absurdum) and negation elimination (Double)
 A: 1) $\big((P\rightarrow Q)\rightarrow P\big)$ --- assumed [a]
2) $\lnot P$ --- assumed [b]
3) $P$ --- assumed [c]
4) $\bot$ --- from 2) and 3)
5) $Q$ --- from 4) by ex falso quodlibet (i.e. : $\bot \vdash \varphi$) or directly from 2) and 3) (skipping 4)) by negation elimination (i.e. : $\varphi, ¬ \varphi \vdash \psi$) 
6) $P \to Q$ --- from 3) and 5) by $\to$-intro, discharging [c]
7) $P$ --- from 1) and 6) by $\to$-elim
8) $\bot$ --- from 2) and 7)
9) $P$ --- from 2) and 8) by double negation elimination, discharging [b]

10) $\vdash \big((P\rightarrow Q)\rightarrow P\big) \rightarrow P$ --- from 1) and 9) by $\to$-intro, discharging [a].


The use of double negation elimination is unavoidable, because Peirce's law is not intuitionistically valid.
A: Assume the antecedent.  Assume the negation of the consequent.  Assume the antecedent of the antecedent of the antecedent.  Assume the negation of the consequent of the antecedent of the antecedent.  Derive a contradiction.  Infer the negation of the negation of the consequent of the antecedent of the antecedent.  Infer the consequent of the antecedent of the antecedent.  Infer the antecedent of the antecedent.  Infer the consequent of the antecedent.  Infer a contradiction.  Infer the negation of the negation of the desired formula.  Infer the desired formula.
Anytime you have a formula like (($\alpha$ $\rightarrow$ $\beta$) $\rightarrow$ $\gamma$), you might see if you can assume $\alpha$ and then derive $\beta$.  Then you can infer ($\alpha$ $\rightarrow$ $\beta$) and then infer $\gamma$.
A: I've just checked Mauro's answer on the forall x proof builder and checker. 
Here are two useful links: summary document, Calgary remix.
Here is a screenshot: 

Here is a LaTeX transcription:
$01\ \square\ ((P\to Q)\to P)$
$02\ \square\ \square\ \lnot P$
$03\ \square\ \square\ \square\ P$
$04\ \square\ \square\ \square\ \bot\qquad\bot$I 2,3
$05\ \square\ \square\ \square\ Q\qquad\bot$E 4
$06\ \square\ \square\ P\to Q\qquad\to$I 3-5
$07\ \square\ \square\ P\qquad\to$E 1,6
$08\ \square\ \square\ \bot\qquad\bot$I 2,7
$09\ \square\ \neg\neg P\qquad\neg$I 2-8
$10\ \square\ P\qquad$DNE 9
$11\ (P\to Q)\to P)\to P\qquad\to$I 1-10
