1. In these circumstances, the common law of contract applies. Under it, a neighbour may offer a thin strip of his property to his next door neighbour by oral offer. there is no need for the offer to be in writing. However, in order to have a valid transfer of the strip of land, the transfer has to be in writing and specify the parcel of land transferred, the price for this transfer and the parties involved. Otherwise, by virtue of the Statute of Frauds, such a transfer will not be valid, if not in writing. So, if you intend to take up the offer of your neighbour to buy the strip of land you will need to write it down in order that it be valid. Additionally, you should register the transfer in the Land Registry.
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3. I would not advise you to date it retrospectively as if a fraud can be shown, then the transfer can be set aside. So, if you want the transfer to be impeachable, use the correct date when the transfer occurred, not some retrospective date. Incorrectly altering the date will lead to problems.