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In the letter, Watson wrote about his reaction to a child sex offender moving into the dormitory.
He believed that the man was taunting other inmates by watching children's TV and said he struggled to sleep "having not done what every instinct told me I should've done right then and there".
Two hours before the fatal attacks, Watson took his concerns to a prison counsellor and urgently requested a transfer back to more secure accommodation "before I really **** one of these dudes up".
His request was dismissed.
"I was mulling it all over when along came Molester #1 and he put his TV right on PBS Kids again," he wrote, according to the newspaper.
"But this time, someone else said something to the effect of 'Is this guy really going to watch this right in front of us?' and I recall saying, 'I got this.' And I picked up the cane and went to work on him."
On the way to hand himself in for the attack, he saw "a known child trafficker, and I figured I'd just do everybody a favour", Watson wrote.
"In for a penny, in for a pound."
Watson is now being held away from other prisoners while an investigation is completed.
In the letter, he defended his actions, writing: "Being a lifer, I'm in a unique position where I sometimes have access to these people and I have so little to lose.
"And trust me, we get it, these people are every parent's worst nightmare."