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March 23rd, 2014, 12:57 PM  #1 
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407  Some binary and ternary facts How many weights are needed to weight integral weights from 1 to 100? We can use seven binary weights [color=beige]. . [/color]to weigh up to 127 pounds. To weigh 100 pounds, convert 100 to binary:[color=beige] .[/color] This means:[color=beige] .[/color] If we can place weights in both pans of the scale, [color=beige]. . [/color]we can use five ternary weights:[color=beige] .[/color] [color=beige]. . [/color]to weigh up to 121 pounds. To weigh 93 pounds, convert to ternary:[color=beige] .[/color] This means:[color=beige] .[/color] It can get a bit tricky. To weigh 100 pounds, convert to ternary:[color=beige] .[/color] This means:[color=beige] .[/color] But we don't have two 9pound weights . . . What can we do? Add a 9pound weight to each side. Of course, we don't have three 9pound weights, either. [color=beige]. . [/color]But we can replace them with a 27pound weight. [color=beige] . . [/color]There! Must we go through this juggling every time? . . . No! Convert 100 to ternary:[color=beige] .[/color] If a "2' occurs, replace it with "1" and add 1 to the digit to the left. The "minus" means that that weight goes on the "other side". 
March 23rd, 2014, 01:39 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Some binary and ternary facts 

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