Novel I read more than 50 years ago about a man with three brains that will wake up successively [duplicate]

Question

He was an average accountant. One day at work, he suddenly feels like throwing up. He goes to the toilets. When he returns, his mathematical abilities are increased tenfold. His second brain has awakened.

Later on, other people (probably) from the future are looking for him. When they capture him, they're going to cut off his third brain before it wakes up, but it's just too late. His third brain wakes up in the surgery room and there, the protagonist masters the atomic composition of his body. He can thus prevent the surgeon's scissors from cutting his third brain, and he can get up despite the bonds and escape by crossing the walls.

Answer 1

This is One Against Time (1970) by Astron del Martia.

The novel is available to borrow at the Internet Archive and the blurb on the back of the book covers a couple of the plot points you mentioned:

• The protagonist, Harold Newman, is a bank clerk whose mind is activated by a warp in time, giving him a 'super brain.'
• People from the future are looking for him, as they perceive him as a threat to the continuance of life on Earth.

Harold Newman was a loser.
He was a plodding bank clerk whose past was weary and whose future was dull.
Until the moment when his mind was activated by a warp in time and he became a super brain—a brilliant freak gifted with the power of twenty computers.
Suddenly Harold's future wasn't dull anymore.
It was deadly.
His very existence was a threat to the continuance of life on Earth. He knew too much. The World President issued an implacable order through the spiraling levels of space and time:
Search the past! Find the super brain and kill him. But do it now!

I took a quick look at the book and there is a scene early on (beginning around page 12) where Harold is struggling with a balance sheet at his workplace, visits the restroom to throw up, then finds the balance sheet much easier to deal with when he returns to his desk.

There's also a later scene (beginning around page 128) where three men are beginning to operate on his brain and he suddenly reaches a higher level of consciousness, enabling him to escape and walk through a wall by filtering the particles in the wall between those in his body:

Newman concentrated until he was conscious of every single electron and neutron in his body and clothing. At every split second he knew the position of every one of those millions of electrons and neutrons, He knew each of them individually and precisely. In the same way he knew the brick wall against which he stood. He was conscious of every one of the millions of whirling electrons and neutrons that gave the brick wall the appearance of solidity.

It needed mental agility and a few seconds to master the system. But it was well within his capabilities. Tentatively he extended a forefinger and thrust it slowly at the brick wall.

His complete consciousness of the whirling electrons and neutrons in his finger was as complete as his knowledge of the electrons and neutrons in the wall. His mind was strong and controlling. It exerted guiding influences over the madly whirling particles. The particles of matter that were the wall met the electrons and neutrons of his finger. His swift, controlling brain guided, influenced and merged them. He. knew a million combinations in the same instant and skillfully filtered the tiny particles of the wall through the tiny particles that constituted his flesh.

When his finger had merged into the brick wall as far as a wrist he had mastered the trick. His arm and body followed with ease as he concentrated on preventing the electrons and neutrons of the brick wall colliding with the electrons and neutrons of his body.

He filtered through the brick wall like water flowing through a sieve. He paused a moment the other side in a lighted corridor, and relieved from the strain and concentration of filtering, knew instantly the direction he must take.