# Smart Weights

A Starter Problem:

A box contains weights of size 1 , 2 , 4 , 8 and 16 ounces.

Show how to use different combinations of these weights to measure any integer weight from 1 ounce upwards.

How far can you get with these 5 weights ?

New Idea:

Traditionally, in a balanced pan weighing machine, the quantity of food to be measured goes in one pan, while the standard weights go in the other, until the two balance. Your calculations have been for this kind of machine.

A small adaptation involves putting a hook beneath the food pan, so that weights may be placed on either side of the balance.

This enables one to use subtraction of weights, as well as addition For example, one could use weights of 1 oz and 3 oz to measure 2 oz of food, by placing the 1 oz with the food, and the 3 oz in the opposite pan. The pans will balance when there is exactly 2 oz of food together with the 1 oz weight, balancing the 3 oz in the other pan.

We can write this as 2 = 3  1 !

So far, so good.

### The Problem Itself:

1. Given weights of 1 , 3 and 9 oz, how many different quantities can you weigh, using the hook method whenever you want ?
1. Look at the sequence in the weights so far, and guess what the next weight to be used should be. Then use your four weights to continue the weighings as far as you can.

Youve probably discovered that you could get further using only four weights, and a hook, than you did with a set of five weights , using the normal 'balance pan' approach.

Interestingly, the advantage becomes more marked the further one wants to take it. So whats stopping us using this method ?!

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