Out of n consecutive natural numbers, four numbers are selected such that the differences of all the possible pairs are distinct. What is the least value of n for which this is possible? Pick one:	A) 6	B) 7	C) 8	D) 9	E) 10
Explanation :   Let the 4 numbers be X1, X2, X3, X4 where X1&lt;X2&lt;X3&lt;X4. Let X1 + a1 = X2, X2 + a2 1 +  1= 
<gadget id="calculator">1 +  1</gadget>
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X2, X2 + a2= X3, X3 + a3 = X4. The difference of X1 and each of the other three, of X2 and each of the greater two, of X3 and X4 are tabulated.   X1	X2	X3   X2	a1   X3	a1+a2	a2   X4	a1+a2+a3	a2+a3	a3   We require that n be minimized and these 6 expressions have distinct values. Let X1 = 1   Now, n = 1+a1+a2+a3. Now a1, a2, a3 can be chosen as the least possible values i.e. 1,2 and 3. However the order of chosing should be such that the six expressions (differences) should all be distinct.   By little trial and error (a1, a2, a3) can be (1, 3, 2) or (2, 3, 1) in which case n = 7 and the four numbers can be (1, 2, 5, 7) or (1, 3, 6, 7) respectively.   Hence the minimum value of n is 7.   Answer : 7
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Out of n consecutive natural numbers, four numbers are selected such that the differences of all the possible pairs are distinct. What is the least value of n for which this is possible?