For Patio.

Each of Paul and Jenny has a whole number of pounds.
He says to her: “If you give me £3, I will have n times as much as you."
She says to him: “If you give me £n, I will have 3 times as much as you."
Given that all these statements are true and that n is a positive integer, what are the possible values for n?
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Okay. Before you read MY details of working, you should know that this may be wrong, and if not, that there are other methods of solving this.
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Let P and J be Paul's and Jenny's pounds respectively.

	P + 3 = n( J - 3 )
	3( P - n ) = J + n
				P - nJ = -3n - 3
				3P - J = 4n
						3P - 3nJ = -9n - 9
						3P - J = 4n
									3nJ - J = 13n + 9
												n( 3J - 13 ) = J + 9

From this:	3J - 13 ≤ J + 9				3J - 13 ≥ 0
		J ≤ 11					J ≥ 5
				5 ≤ J ≤ 11

Through trial and error, the only solutions I get are:

	n = 1	J = 11			n( 33 - 13) = 9 + 11
	n = 2	J = 7			n( 21 - 13) = 9 + 7
	n = 3	J = 6			n( 18 - 13) = 9 + 6
	n = 7	J = 5			n( 15 - 13) = 9 + 5

Therefore the solutions are n = 1, 2, 3, 7

-Marko Polo