Cryptology - Unofficial St. Mary's College of California Web Site

220 CHAPTER 11. KNAPSACK CIPHERS
Example: There are 8 piles of gold bars. Every bar in a given pile weighs
the same. Each in pile 1 weighs 18oz. In pile 2 each weighs 29oz. Pile 3: 34oz,
Pile 4: 41oz, Pile 5: 67oz, Pile 6: 88oz, Pile 7: 101oz and Pile 8: 119oz. Your
backpack can carry a maximum of 300 oz (about 19 lbs). Which bars and how
many should you pick 1
⋄
11.2 A Related Knapsack Problem
An apparent simplification is to assume the cave contains exactly one bar of
any particular weight. So you only need to decide “yes” or “no” to each bar,
rather than deciding how many bars from each pile to take. We also modify the
problem to demand an exact amount of gold.
Examples:
(1) The weights of the bars are 4, 7, 12, 19, 22, and 25 lbs. If the total amount
demanded is 55 lbs., this weight can be supplied: 4 + 7 + 19 + 25 = 55.
However, there is no way to choose bars of total weight exactly 50 pounds.
(2) The weights of the bars are 27, 31, 41, 48, 55, 59, 62, 65, 73, and 77. Total
amount wanted is 257. Is this weight possible What about 364 2
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
This problem may seem easier than the first, but it is still quite hard. Why
Think of the number of possibilities that must be considered. For each bar we
must decide whether we want it or not:
For one weight, say 27 lbs.: Do we want it yes or no (2 = 2 1 choices).
For 27 and 31: yes/no and yes/no, or yy, yn, ny or nn (4 = 2 2 choices).
For 27, 31, and 41: y/n and y/n and y/n gives yyy, yyn, yny, nyy, ynn,
nyn, nny, and nnn (8 = 2 3 choices).
By the time we have 5 weights we must make 2 5 = 32 choices, 8 weights demand
2 8 = 256, and 10 weights, as in the last example, leads to 2 10 = 1024
possibilities, too many for a quick answer if checking by hand. In fact, if there
were 100 bars to choose from (a moderately rich dragon, in other words), there
would be
2 100 ≈ 1, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000
1 I can make 295 as 2 × 18 + 2 × 34 + 3 × 41 + 67. Can you do better
2 255 is possible (in seven different ways!) but 364 is not.