Instructions

This surprising illusion is inspired by other classic "geometric vanishing" puzzles. (You can read more about these in Martin Gardner's Mathematics, Magic and Mystery.) Depending on the age of your child, this activity will take a lot of "working together." Even if you end up doing most of the work, you can still do a lot together: counting as you measure, coloring, decorating, and of course counting the candles of each color in disbelief!

Use a ruler to draw a line down the center of a standard letter-size sheet of paper (8.5"x11").

Now adjust the ruler so that the left edge of the paper aligns with the 1/2" mark on the ruler.

Starting at 1", make small tick marks 1/8" on either side of the inch marker. Repeat this for each inch marker from 1 to 11. This will give you room for 11 candles.

Before you move the ruler, also make a tick mark at 4.5". (This is where you'll cut the paper later.)

Now it's time to draw your candles. Pick any two colors. We used green and blue. Draw green candles at positions 1, 4, 8, and 11. It helps the illusion if the candles are different heights. Be sure there is at least some candle above and below the center line. Draw blue candles at positions 2, 3, 9, and 10. For now, these "candles" are nothing more than vertical bars. The decorating comes later.

In positions 5 and 7, draw two more green candles, one completely below the line, and the other completely above it. Then draw one more blue candle (above and below the center line) in the middle.

Now cut at the 4.5" mark (between candles 4 and 5) and also along the center line.

Decorate the candles by coloring them in, and adding wicks, flames, bases, and drops of wax. Be sure that the flame for candle 5 is above the cut, and the base for candle 7 is below the cut.

Now get ready for the magic! One arrangement of the bottom pieces gives 6 green candles and 5 blue. The other arrangement gives 5 green and 6 blue!

Variations d

Fewer (or More) Candles:

Try making versions with other numbers of candles. Here is a 5-candle version:

Other Pictures:

The illusion works best with any object that can be either short or tall, and that has distinct "top" and "bottom" pieces. Here is a version with 13 pencils. What else can you think of that might work?

Learning Notes d

Understanding the Illusion:

This and similar illusions rely on what mathematics writer Martin Gardner called the "principle of concealed distribution." Take a look at this simplified version of the trick, in which 9 lines seem to turn into 8.

Can your child explain where the "missing" line went? Consider numbering the lines at the top and at the bottom. Then slide the top over. This shows "where" the missing line actually went. All of the original line 1 (which was entirely above the cut) combined with a small part of line 2. The rest of line 2 combined with some of line 3, and so on, until the end when we see that all of line 9 (which was entirely below the cut) combined with a small part of line 8.

You can also ask how the combined length of every line in the first picture would compare to the combined length of every line in the second picture. (Kids might even try recreating the lines on their own paper, then measuring.) The combined lengths would be the same, because no length was actually removed. Rather, a small length of each line was distributed among the other lines.

To better conceal the trick, the extra bit of paper on the lower left could be cut off and used to fill the gap on the lower right. That is essentially what the candle illusion does with its two interchangeable pieces.

In the candle illusion, two different sets of lines have been interspersed to give the impression that one candle is changing color. Here are the green and blue candles separately.