This sculpture/puzzle is my G4G11
exchange item. It is a puzzle for you to assemble and
makes an interesting 10 cm cube sculpture. It should be
displayed standing on a corner as shown, 15 cm tall. Be
warned that it is trickier than it looks and requires some care
to avoid ripping or damaging the cards. I have pre-cut slits in
each card with the proper lengths and angles for them to hold
When properly assembled, there
are six square tunnels leading to the center, like the one
shown above. Each square tunnel starts in the center
of a face, passes through the center of the cube, and comes
out in the opposite face. The sides of the tunnel are
rotated 45 degrees relative to the sides of the cube.
There are also eight triangular tunnels leading to the
center, like the one shown above. Each starts in a
corner, goes through the center of the cube, and continues
out to the opposite corner. So the center of the cube
is a complex arrangement of intersecting tunnels, with
fourteen ways to exit.
The puzzle is assembled from twelve playing cards.
I've made packages of thirteen identical pre-slit cards. You
can see what card is in your pack by looking on the back of
the bag. All thirteen cards in a pack are the same, e.g.,
all the Ace of Hearts. You only need twelve cards. The
thirteenth is a spare in case of catastrophe.
Step 1 is to carefully fold each card along the partially cut diagonal, as shown
rip anything. The picture side is inside, leaving
the back visible from both sides. A thin metal ruler makes
this easy. Just line the
diagonal over the edge and push down with your thumb. Crease
well, then let it spring open naturally to roughly a 60
degree angle. (If you happen to have a table or counter
with a sharp edge, that also works well for creasing.) No
additional bending or creasing is necessary.
The twelve cards are all identical, with the same slots
and the same crease.
Step 2 is to carefully slide the end of
one card on to the two short slots in the side of
another card, as shown above. It takes only a slight
flex of the card, then it locks into place. It is easier
if you first connect the diagonal slot, which is
slightly longer. The lengths work out so the edges of
the two cards cross as shown. Keep in mind how basic
this is if you get frustrated later, because every
connection in this puzzle is exactly like this.
Step 3 is where
people sometimes make a mistake, so look carefully at the image
above. First notice that two corners of each card
have a digit or letter while the other two corners have a
diagonal slot. Then observe how two of the digit (or
letter) corners are near each other (the A's at the
top of this image). You want to add a third card (the one
in back in this image) to join each of the first two in a kind of 3-way cycle,
so that the backs of the three digit corners make a
triangular tunnel like the one at right in the image
Step 4 Make a second three-card
module, just like the first. The image above
shows two views of the identical three-card structure.
Step 5 Join the two
three-part modules together to create a square tunnel like
the one above. You need to make two connections to
compete the square.
Step 6At this stage, you have completed one square
face of the cube and parts of the four surrounding
faces. Look at the spiral structure of the completed
square and visualize how there will be an identical square
tunnel in the centers of the surrounding faces. Add
the remaining six cards one at a time, being
sure to make the square tunnels 4-sided and not
3-sided. Inserting the last card is tricky, because
four connections need to be made, so be patient and be
careful not to bend or rip the cards.
Donít get upset if it tends to fall apart during
construction. The cards are slippery, but everything holds
together securely when the twelfth piece is
inserted. When done, look on all sides and make sure
all the slots are completely engaged.
If you want
to make more of these Tunnel Cubes, you can print out
the above template and copy it on to your own cards.
Then cut out all ten slots in each card. Be sure
the cards are face-up as shown, so the slots on the
diagonal do not go through the letter or digit
of the card value. The dotted line indicates where
If you want to understand what you built and think about
the underlying mathematical ideas, you might first find
all the symmetry axes. The 4-fold and 3-fold axes
are easy, because they go through the tunnels, but can
you find the six 2-fold symmetry axes? Then think
about the set of planes in which the cards lie.
Twelve folded cards could define twenty-four planes in
space, but in fact there are only twelve planes, because
each triangular half of any card is coplanar with
another half card. What polyhedral shape defines
these twelve planes? (It is the intersection of
twelve infinite half-spaces defined by these
planes.) Notice the twelve planes come in six
parallel pairs, so there are only six normal directions;
characterize them with respect to the cube. The
cards automatically adjust to make everything work out
without you having to know the exact fold angle; what is
the fold angle?