After your rise to fame as—well, whatever you're famous for—you've decided to hire a team of bodyguards. They have agreed to payment in the form of a rectangular prism gold bar that you possess. Your agreement states that you will pay them 1/7 of the gold bar every week for 7 weeks.

Obviously, you could make six cuts to divide your gold bar into sevenths so that you could pay your weekly 1/7 fee, but there is a better way that involves fewer cuts.

Make a plan. How many cuts should you make? How will you pay the bodyguard team each day?

Solution

It's possible to do this with only two cuts. You start by cutting the bar into three pieces that we'll call A, B, and C such that: A = 1/7 of the bar; B = 2/7 of the bar; and C = 4/7 of the bar.

Here's the payment plan: Day 1: Pay them with piece A; Day 2: Pay them with B but get back A; Day 3: Give them A; Day 4: Give them C but get back A and B; Day 5: Give them A; Day 6: Give them B but get back A; Day 7: Give them A.

This is really a binary system in which with the ones, twos, and fours columns, you can make any number up to 7.

What if it were 15 days and each day required 1/15 of the bar? What other number of days would work this way?

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