Teacher Jane Longhurst's body was kept in a cellar or a fridge for more than a month before she was dumped and burnt, police believe.
Detectives said she was strangled soon after her disappearance on March 14 and her killer kept her body concealed somewhere cool until the Easter weekend.
Her body was starting to decompose and, fearing neighbours might become suspicious, the killer was "spooked" into dumping her and trying to destroy evidence.
The murderer, who could well live in the Brighton and Hove area, must have read and watched news reports of the search for five weeks while the teacher's body lay perhaps yards away.
Police are asking what kind of person could commit such a heinous crime and carry on living normally without showing signs of stress or trauma.
They think someone must have spotted changes in the killer's behaviour and are urging them to come forward.
Jane's burning body was discovered at Wiggonholt Common, near Pulborough, on Saturday evening.
A passing driver noticed smoke at 8.17pm and alerted the fire brigade, fearing a woodland blaze.
The body was burnt beyond recognition but checks on dental records confirmed it as Jane. It was a tragic breakthrough for the 60-strong police team who had been searching for her.
Detective Chief Inspector Steve Dennis, heading Operation Keen, said last night he could not be drawn on a possible motive but detectives have their private theories.
There was no indication Jane was having a secret affair - and it would have been completely out of character - but it was possible she had a secret admirer.
Did he meet or bump into her that day and lose his temper when she rejected his advances?
One Sussex Police insider said: "There are not that many possibilities but that is certainly one. The worst-case scenario is that this was a planned abduction-murder by a stranger."
Mr Dennis said dumping Jane's body at a popular, busy beauty spot was an act of panic.
He said: "That person may have come home smelling of petrol or smoke. If anyone out there knows someone whose behaviour has become bizarre then they should contact us."
Mr Dennis said it was too early to say whether Jane had been sexually assaulted. He said her body, though decomposed, had been reasonably well preserved "somewhere cool".
Jane, 31, a teacher at Uplands special school in Hollingdean, Brighton, was last seen by a neighbour at 2.40pm on March 14 walking to her front door, carrying two shopping bags.
Her partner Malcolm Sentance returned to their address in Shaftesbury Road, Brighton, just after 4pm that day and became increasingly concerned as the evening wore on.
He telephoned friends and relatives and at midnight alerted police.
There has been no trace of the shopping bags or Jane's black leather Next purse, her blue-grey Nokia mobile phone, her flat keys and some items of clothing including a denim jacket and red suede shoes.
Police are keen to speak to the drivers of three vehicles seen near Wiggonholt Common on Saturday - a Renault people carrier, a Volkswagen and a Ford Focus.
Anyone with any information should call the Operation Keen incident room in Brighton on 0845 6070999.
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