175PUZZLE PLAY 2001 AIMS Education Foundation

Puzzle TopicVisual paradoxes

Puzzle QuestionHow do you explain the bear which seems to disappear and thenreappear when you rearrange the pieces of this puzzle?

MaterialsStudent sheet (run on cardstock)ScissorsColored pencils, optional

Puzzle BackgroundBearly There is an almost magical puzzle that is sure to

fascinate students and teachers alike. It is patterned after theintriguing disappearing puzzles that were the rage in the late 19thand early 20th century. In these puzzles, pieces are rearranged orrealigned in such a way that one or more of the objects pictureddisappears or appears.

The puzzle presented here has three pieces: a long bottompiece and two shorter, top pieces. In one arrangement of thesepieces there are 14 bears, in another, 13. The challenge facingstudents in this puzzle is to explain how this visual paradox works.

While it may not look very mathematical in nature, this puzzleis full of interesting mathematics. In fact, one of the bestexplanations of the disappearing and appearing phenomenon isthe mathematical one (see Solutions).

176PUZZLE PLAY 2001 AIMS Education Foundation

Puzzle Presentation1. There are two pages included for this activity which should be

run on cardstock or some similar material. The first page hastwo small copies of the puzzle which can be cut out for studentsto use. The second page has one larger copy of the puzzlewhich can be assembled and laminated for use in a center orthe classroom puzzle corner.

2. The first student page can be shared by two students since ithas two copies of the puzzle. However, you may want to giveeach student a copy of this sheet. This way students will havetwo copies of the puzzle; a clean copy and one they can markup (this helps when trying to figure out how the puzzle works).

3. This puzzle should not have any imposed time limits. The longerstudents work with the puzzle, the more they will discover. Fromtime to time it may be appropriate to have students share thediscoveries they have made up to that point.

4. This is a great puzzle to send home with students. Because it isso intriguing, parents will likely be as captivated by the puzzleas their children. This helps foster a positive school-to-homelink.

Solution HintsCount the pieces of bears on the three strips. Measure the bears

in both arrangements and compare their average heights.

177PUZZLE PLAY 2001 AIMS Education Foundation

How

man

y be

ars

are

on e

ach

strip

bel

ow?

Care

fully

cut

one

of th

est

rips

apar

t al

ong

the

lines

. Sw

itch

the

top

two

piec

es a

nd c

ount

the

bear

s ag

ain.

Wha

t do

you

not

ice?

How

doe

s th

is h

appe

n?

178PUZZLE PLAY 2001 AIMS Education Foundation

Teac

her's

Pat

tern

:1.

Cut

out

all f

our

strip

s.2.

Tape

the

mar

ked

edge

s to

geth

er. Th

isfo

rms

the

bott

om s

trip

of th

e pu

zzle

.3.A

ltern

ate

the

top

strip

s an

d co

unt

the

bear

s ea

ch t

ime.

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