Question

Dear Mitch,

When are to going to post the solution to that high school level Halloween challenge problem with the skulls, monster heads, bats and pumpkins on the balance scales?

I'd appreciate it if you could put it up a couple of days before the holiday so I'll be able to go over it with my class.

- M.J.K.

Answer

Dear M.J.K.,

Now!

To quickly review, four balance scales were shown, scale A, scale B, scale C, and scale D. 

Scale A, scale B, and scale C were balanced, but the fourth scale, scale D, was not yet balanced.

Scale A depicted: 1 skull and 1 Frankensten head on the left-hand side; 1 bat on the right-hand side.

Scale B depicted: 1 skull on the left-hand side; 1 Frankenstein head and 1 pumpkin on the right-hand side.

Scale C depicted:  2 bats on the left-hand side; 3 pumpkins on the right-hand side.

Scale D, which was NOT yet balanced, depicted: 1 skull on the left-hand side; ? on the empty right-hand side.

The question was:  How many "Frankenstein heads" would one need to place on the empty side of scale D in order to balance out the right-hand side of that scale, which held just 1 skull?

SOLUTION:

5 Frankenstein heads.

HOW?

Look: In order to make the 2 left-hand sides of scales A and B equal in weight (so that they will balance on a balance scale), add 1 Frankenstein head to each side of scale B.

This means that 1 bat = 2 Frankenstein heads + 1 pumpkin (OR). . . 

2 bats = 4 Frankenstein heads + 2 pumpkins.

Scale C also shows that 2 bats also equals 3 pumpkins.

(OR) . . .

4 Frankenstein heads + 2 pumpkins = 3 pumpkins.

Now, subtracting 2 pumpkins from each side, we find that the weight of 4 Frankenstein heads equals the weight of 1 pumpkin.

Next, we exchange the 1 pumpkin in scale B with 4 Frankenstein heads. . .

and so we can . . .

BALANCE THE SKULL IN SCALE D WITH 5 FRANKENSTEIN HEADS!! 

SUGGESTION:  Since this particular problem seems to present more of a challenge to most students than many of these types of 'balance-scale' problems, it can be very helpful to make a bunch of paper cut-outs of each of the four variables (the Frankenstein heads, the skulls, the pumpkins, and the bats) and, using a bit of scotch tape or other form of adhesive, have students try moving the pieces around on various 'scales', which you could simply draw on the board, enabling them to benefit from the hands-on aspect of such problem-solving exercises. Some teachers have told me that they find the best system is to have different colored Post-its, which can, of course, be easily moved from position to position.

Enjoy!

(Oh, and Happy Holiday!)

- Mitch