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Balance Point

Age 14 to 16 Challenge Level:
Well done to Jim sent us his solution:

The trick to solving this problem is to work systemticlly. The 8 weight can be placed on either the end or the inner point, and it is symetrical so we only have to look at on side.

If the 8 is on the end the arrangements to consider are:

If the 8 is on an inner attachment point we need to consider:

I them tried to find the balance points, and found they were in the central interval except for 8241, 8412 and 8412. I found that 4821 and 8142 produce a balance point at one of the inner attachment points.

For 8241, and 8412 the bar needs weight 4/3, for 8421, 8/3 is required. So the bar must be at least 8/3 in weight for all arrangments to have balance points in the central interval.