Brain-Teaser 418: Bell’s weights

From The Sunday Times, 11th May 1969 [link]

“Now”, says Bell at the pub, “look intelligent and imaginative even if you don’t look beautiful by any means”. We swallow the insult and look solemn. Bell likes his jokes and we like his puzzles.

“Imagine you have some scales, but only three weights. However, you have a heap of Grade A sand, and a couple of bags; so you make two extra weights, one as heavy as all the first three put together, and the other twice as heavy as all the first three. Whereupon all the remaining sand is removed to a great distance”.

“With these five weights you must be able to weigh 1 ounce, 2 ounces, 3, 4, and so on, as far as possible. No gaps in that lot. It’s how far you can to the first gap I’m after. Usual prize — one pint for the best score before closing time”.

What should Bell’s five weights be to give the highest possible score?

This puzzle was included in the book Sunday Times Brain Teasers (1974, edited by Ronald Postill).