Take the number 1. Double it. Then double it again. If you continue doubling, you’ll get a series of numbers called “powers of 2”--a product of a bunch of 2s. (This may seem strange, but think of the number 1 as a product of no 2s. Remember, every number has 1 as a factor.)
For example, 16 is the product of four 2s. If you reverse the process, you divide by 2 each time. So 8 is the product of three 2s, 4 is the product of two 2s, 2 is the “product” of only one 2, and then 1 is the product of no 2s.
An interesting thing about powers of 2 is that every counting number can be written as a sum of these numbers.

1. Try it with some small numbers. Write the first eight counting numbers (1-8) as sums of powers of 2. Remember, these are 1, 2, 4, 8, 16, . . . .
2. In this problem, you’ll see why every counting number can be written as a sum of powers of 2.
   1. Suppose you have balance scales and a set of weights that are powers of 2: 1 gram, 2 g, and 4 g. Show that you can accurately weigh any object whose weight is a counting number from 1 g to 7 g.
      With balance scales, you put the object to be weighed on one side, then you add weights to the other side until the two sides balance. The weights will be the same on the two sides.
      
   2. Now suppose someone gives you an 8 g weight for your scales. Can you still weigh 1 g to 7 g? What other weights can you find?
   3. What additional weights can you find if you also get a 16 g weight?
   4. What does adding the next power-of-2 weight do to the possible weights you can find?
   5. Suppose you get a set with a large number of pieces. Describe how to calculate the largest weight you can find, just by knowing how many pieces there are.
3. Try to find the quickest way to weigh an object using the power-of-2 weights.
4. Use your answer to problem 3 to write 227 as a sum of powers of 2.

1. 1 = 1, 2 = 2, 3 = 2 + 1, 4 = 4, 5 = 4 + 1, 6 = 4 + 2, 7 = 4 + 2 + 1, 8 = 8
2. 1. For 1 g to 7 g, choose weights as given in the answer to problem 1--for example, 3 g can be weighed as 1 g and 2 g.
   2. Yes, you can still weigh 1 g to 7 g as before. With the added weight, you can also get 8 g to 15 g, just by adding the 8 g weight to the 1 g to 7 g combinations.
   3. Adding a 16 g weight allows 16 g to 31 g.
   4. It doubles the number you can get, adding weights from the weight of the new one to the sum of all the previous ones. (This sum is one less than the new weight, so you can get up to one less than twice the new weight.)
   5. Multiply as many 2s as you have pieces, and subtract 1. You can get up to that result for possible weights. (For example, a six-piece set could go from 1 g to 2 × 2 × 2 × 2 × 2 × 2 - 1 or 31 g.)
3. Start with the heaviest weight that isn’t heavier than the object. Then add the next heaviest weight that won’t make the total heavier than the object. Continue adding the next heaviest weights that aren’t too heavy, until you get the correct weight.
4. The powers of 2 are 1, 2, 4, 8, 16, 32, 64, 128, and 256. (The next one is larger than 277.) So start with 256. Using the weight context, you see that adding any weight heavier than 16 would add too much. With the 256 and 16 weights, the total is 272. You can either recognize that the remaining 5 can be added as 4 and 1, or continue as before: adding 8 adds to much, but adding 4 does not, giving 276. The 2 weight adds too much, and adding 1 gives 277. So 277=256+16+4+1.