Tanton’s Two-Pan Balance Puzzle

I’m a big fan of pan-balance puzzles in which you’re given a two-pan balance and asked to use it to uncover a counterfeit coin or determine the weight of a coin. One classic example is the following puzzle:

You have 12 coins that all look exactly the same. One is counterfeit and is either heavier or lighter than the other 11. With a two-pan balance, isolate the counterfeit coin in three weighings.

A much simpler pan-balance puzzle asks how to weight any object from 1 to 31 kg using the powers of two as weights: 1, 2, 4, 8, 16 kg. I used this idea as the basis for my Arranging Addends puzzle.

Below is a new pan-balance puzzle from mathematician James Tanton’s Twitter feed on April 11. Tanton punctuates the puzzle with an exclamation point, and rightly so: It’s beyond amazing that with just five different weights, you can determine the weight of any rock from 1 to 243 kg. Stop reading here if you’d like to attempt the puzzle.

Spoiler alert! The weights of the five rocks are 2, 6, 18, 54, and 162 kg.

While this puzzle is almost certainly too hard for children to solve, it does serve as the basis for an excellent puzzle where children can apply their logical reasoning skills. In the websketch below (and here), there are  two rocks with known weights—2 and 6 kg—and a rock with an unknown weight between 1 and 9 kg  sitting on the left pan of the balance. Can you  determine its weight by using the pan and the rocks?

On page 2 of the websketch,  the weight of the mystery rock is between 1 and 27 kg and you now have three rocks with known weights—2 , 6 and 18 kg—to determine the unknown weight.

The final page of the websketch features the full set of five rocks (2, 6, 18, 54, and 162 kg), but frankly, I think these challenges are a bit much!

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