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Candle Conundrum

Create an illusion sure to amaze! Watch as one candle magically changes color before your eyes.

Ages: 4+ (with help)
Players: 1
Time: 20+ Minutes
Type: curiosities
Location: tabletop
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Ages: 4+ (with help)
Players: 1
Time: 20+ Minutes
Type: curiosities
Location: tabletop

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").

Tracing a line across the center of a paper with a ruler

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

A ruler lined up with the edge of a paper at the half inch mark

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.

Marking along a ruler at an eighth above and below each inch mark

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

Marking the paper at the ruler's four and a half inch mark

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.

Four green rectangles drawn on a paper
Four green and four blue rectangles drawn on a paper

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.

Two more green rectangles added
And one more blue rectangle added

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

The paper cut along the middle and at the four and a half inch mark

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!

Six green candles and five blue candles
Five green candles and six blue candles

Don't forget: it's Beast Academy Playground, not Beast Academy Study Hall. Change the rules, be silly, make mistakes, and try again. The Variations and Learning Notes are here for you if you want to dive deeper, but not all of them apply to learners of every age. The most important thing is to have fun.

Variations d

Fewer (or More) Candles:

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

Three blue candles and two green
3 blue and 2 green
Two blue candles and three green
2 blue and 3 green

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?

Thirteen pencils drawn on paper

Classroom Tips d

Candle Conundrum is a great art activity for practicing measurement with a ruler. It's also just a fun craft for the last day before a break. Let students be creative with their own variations!

Discussion Questions

  • Where does the missing green candle go?
  • Where did the new blue candle come from?
  • Does it matter how many total candles you start with?
  • Does this illusion work when you start with an even number of total candles?
  • Why do the precise measurements with fractions matter?
  • What variations can you come up with?

Alignment with Beast Academy Curriculum

  • Level 2, Chapter 7: Measurement
  • Level 2, Chapter 9: Odds & Evens
  • Level 3, Chapter 10: Fractions

See Variations and Learning Notes for more ideas on how to adapt this activity and incorporate it into your classroom.

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.

Nine lines drawn on a piece of paper
9 lines
Eight lines drawn on a piece of paper
8 lines

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.

Nine numbered lines drawn on a piece of paper
Eight numbered lines drawn on a piece of paper

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.

Six green rectangles on a piece of paper
6 green candles
Five green rectangles on a piece of paper
5 green candles
Five blue rectangles on a piece of paper
5 blue candles
Six blue rectangles on a piece of paper
6 blue candles

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Materials
  • paper
  • pencil
  • markers
  • scissors
  • ruler
Learning Goals
  • wonder
  • counting
  • measuring
  • spatial reasoning
Common Core Standards
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Image of Ms. Q

Ready to level up?

Keep problem solving with Beast Academy’s full math curriculum for students ages 7–13. Check out our captivating comic book series and immersive online platform.

LEARN MORE

Bring problem-solving to your classroom

Keep your entire class engaged with a full book and online math curriculum, for students ages 7–13. 98% of teachers say they’re satisfied with Beast Academy.

LEARN MORE
Image of a BA book