Who needs algebra anyway?

These five balance and weighing puzzles are intended to be solved using only manipulatives. Some of them are suitable for a puzzle-based math fair. They are fun to solve and require a lot of mathematical thinking.

No algebra is needed, but if you have algebra at your disposal, it is interesting to see how each puzzle can be modelled as a system of linear equations.

#1
There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

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How many blue boxes are needed to balance one red box?

So, the solution for puzzle #1 is not difficult: The balance at the top left says that two yellow boxes are the same weight as three blue boxes. So, attending to the balance at the top right, replace the two yellow boxes with three blue boxes to get

By removing three blue boxes from each pan (which  maintains the balance) we see that five blue boxes will balance one red box.


Here's one that is a little more interesting:

#2
There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

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How many blue boxes are needed to balance one red box?

And one that is more challenging:

#3
There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

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How many blue boxes are needed to balance one red box?

Or this one:

#4
There are three different types of boxes. All boxes with the same colour weigh the same.

 These boxes balance.

---

How many blue boxes are needed to balance one red box?

Here is one that is about inequalities.

#5
There are three different types of boxes. All boxes with the same colour weigh the same.

These boxes balance.

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These boxes will not balance.
Which side of the balance will tilt down?

Puzzle #5 is from Sam Loyd, where it was not given as a balance and weighing puzzle, but was set up as a "Tug o' war" problem:

In a tug of war contest, four stout boys could tug just as strong as five plump sisters. Two plump sisters and one stout boy could hold their own against two slim twins. If three plump sisters and the slim twins team up against one plump sister and four stout boys, which side will win?

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There are major differences between using manipulatives and using algebra to solve these particular puzzles.

1. There are no negative numbers when using manipulatives. In the following situation, you cannot isolate the red box by removing the yellow box from the left pan because there is no balancing yellow box that can be removed from the other pan.

Whereas with algebra, we readily subtract y from both sides of the equation

r + y = 5b 

                         to get

r = 5b – y.

2. There are no fractions when using manipulatives. In puzzle #4, I did not ask the question

How many blue boxes are needed to balance one yellow box? 

because there is no answering it using manipulatives alone.

It could be deduced that two and one-half blue boxes are needed to balance one yellow box, but you cannot get this via manipulatives unless your manipulatives are sliced into smaller pieces beforehand. (And you have to use at least a minimal amount of algebra to determine the size of the slices.)

If we use algebra to solve the puzzle, we fairly quickly obtain the fact that

 y =  ⁵⁄₂ b,

even 'though you can argue that this is of dubious meaning in the context of the physical puzzle.

A more complete answer to the question in the post title:

Sometimes you really don't need algebra, but algebra can make thinking a lot easier.