Step 0 is a full square. Step 1 removes the centre 1/9th, leaving 8 small squares. Step 2 repeats the same removal on each of those 8 squares, leaving 64 tiny squares and a self-similar pattern of holes within holes.

Yes — both quantities follow clean multiplicative/additive rules:

• The number of remaining squares is multiplied by 8 at every step, since each surviving square breaks into 8 new smaller squares.
• The number of holes grows by adding the previous step’s square count, because every remaining square contributes exactly one new hole, while old holes simply stay.

Step 0 is one filled equilateral triangle. Step 1 joins the midpoints of the sides, splitting it into 4 smaller triangles, and removes the upside-down centre one — leaving 3 triangles. Step 2 repeats this removal on each of those 3, leaving 9 smaller triangles.

Step 0 is an equilateral triangle. At Step 1, every side is divided into 3 equal parts, and the middle third is replaced with a “bump” (two sides of a small equilateral triangle) — turning the triangle into a six-pointed star. Step 2 applies the same bump-replacement to every one of the 12 sides created in Step 1.

Cutting along lines joining midpoints of adjacent sides removes 4 small corner triangles, leaving a smaller square in the middle (rotated 45° relative to the original, with vertices at the midpoints of the original sides — its area is exactly half the original square).

The 4 removed corner triangles are right-angled isosceles triangles. Any two of them can be joined along their hypotenuses to make a small square; doing this with both pairs gives two small squares — together these two small squares have exactly the same total area as the central square, and in fact the four corner triangles can be rearranged to exactly tile another square equal in area to the central one.

Fold the four side flaps of the cross-shaped net upward to form the four walls of the cube, then fold the top square down to close the lid. All six squares of the cross become the six faces of a cube, and every edge of the net becomes an edge of the cube where two faces meet.

Only the arrangements that show 4 equilateral triangles correctly joined so they fold without overlapping (the big-triangle-of-4-small-triangles, and the “strip-with-one-offset” arrangement) work as nets. A regular tetrahedron has exactly 2 possible nets (up to rotation/flip) — no other arrangement of 4 equilateral triangles will fold up into a closed tetrahedron without gaps or overlaps.

(ii) fails because when folded, two squares end up overlapping on the same face while one face is left uncovered — the row of 4 squares with two flaps positioned awkwardly causes a clash. All the others fold up into a perfect closed cube with each of the 6 squares becoming a distinct face.

The 11 distinct nets of a cube fall into these families based on how the 6 squares are arranged in rows:

• 6 nets with a row of 4 squares, with the remaining 2 squares attached on either side (one above, one below) in various positions.
• 3 nets with a row of 3 squares, and the remaining 3 attached as a row of 2 + 1, in different staggered positions.
• 1 net in a 2×3 staircase (“S”/”Z”) arrangement.
• 1 net in a “zig-zag” arrangement of single squares branching off alternately.

A cuboid net always has 6 rectangles: 2 of each of the 3 different face-sizes (one pair for each pair of opposite faces), arranged in a cross or row pattern with matching edges touching.

Unfold the net so that the top face and the side face the ant sits on lie flat, edge-to-edge, in the same plane. The shortest path is then simply the straight line joining the ant’s position to the laddu’s position on this unfolded net — when refolded, this straight line becomes the shortest surface path on the actual cuboid, crossing the shared edge between the two faces.

The same idea applies: unfold the two relevant faces flat into one plane, mark the ant’s and laddu’s positions on this net, and draw the straight line between them. That straight line — when the net is folded back up — gives the shortest path. Since the laddu sits exactly on the shared edge here, the straight line on the net crosses that edge at the laddu’s own location.

Only the red path is shortest. When the cuboid is unfolded into its net, the red path becomes a single straight line between the ant and the laddu — and a straight line is always the shortest path between two points on a plane. The blue path, when unfolded, bends at the edge crossing and does not form a straight line on the net, so it is longer than necessary.

Depending on orientation, the projection of a square can be:

• A square of the same size — when the square is parallel to the plane.
• A rectangle (narrower than the original) — when the square is tilted about an axis parallel to one of its sides.
• A parallelogram — when tilted in a more general way.
• A line segment — in the extreme case where the square is seen edge-on (perpendicular to the plane).

The projection of a parallelogram is always a parallelogram (or in extreme cases, a line segment) — it can never become a non-parallelogram quadrilateral such as a trapezium or kite.

The projection of a regular \(n\)-gon is, in general, an \(n\)-sided polygon (though typically not regular anymore) — since the projection of each side is a line segment, and the projection of the whole polygon is built up from the projections of its \(n\) sides, joined the same way as in the original. Side lengths shrink unevenly depending on each side’s angle to the viewing direction, so the polygon usually becomes irregular under projection — though it stays a closed \(n\)-sided shape. In special orientations (where the polygon’s plane is parallel to the projection plane), the projection remains a perfectly regular \(n\)-gon, just possibly scaled.

A pyramid with the same triangular silhouette (e.g. a square pyramid, viewed face-on so its outline matches the cone’s triangular profile) gives an identical projection — since projection only captures the outer outline, not the internal curvature. Many different solids — a cone, a pyramid, even an irregular blob with the same outline — can share the same projection.

Yes. For any line segment, if you know its front view, top view, and side view lengths, the actual 3D length of the segment can be recovered using the 3-dimensional Baudhāyana (Pythagoras) relation:

More precisely, each view captures the line’s extent along 2 of the 3 axes (length, depth, height), and combining the projections along all 3 axes via Pythagoras’ theorem recovers the true length. A line parallel to one axis will show its full length in two of the views and shrink to a point/dot in the third.

For any cube combination, the method is the same:

• Top view: look straight down — draw the outline of the “footprint” the combination makes on the ground.
• Front view: look from the front — draw the silhouette ignoring depth.
• Side view: look from the side — draw the silhouette ignoring length (left-right extent).

Apply this to each of the six L-shaped / staircase cube combinations shown — for each one, count how many cube-widths the shape spans along each of the 3 axes (length, depth, height) and sketch the matching rectangle/L-shape for each view.

(i) A shape that reads as a specific letter from the front typically reads as a plain rectangle from the side (since the side view collapses all the depth-wise layers into one flat silhouette) and as a line or thin rectangle from the top (if the letter shape is only built in a single vertical sheet).

(ii) To also get a specific top view, you must add cubes that extend backward (in depth) in just the right footprint pattern, while keeping the front profile unchanged — i.e., duplicate the front letter shape through several depth-layers selectively.

(iii) To control all three views (front, top, and side) simultaneously, you generally need significantly more cubes, since you must satisfy three independent silhouette constraints at once — this is only possible if the three target letter-shapes are mutually consistent (no single small cube can be required by one view and forbidden by another).

(iv) Other simple letter combinations that work well together include L, T, I, H shapes — letters built from straight, blocky strokes are easiest to combine since their footprints are simple rectangles.

For each set of three views, build the solid using this reliable method:

1. Start with the top view — this tells you the footprint (which grid squares on the ground have at least one cube).
2. Use the front view to determine the height of cubes in each footprint column (going left-right).
3. Use the side view to check/adjust the height of cubes in each footprint row (going front-back) — resolving any column where front and side views disagree by adding the minimum number of cubes needed to satisfy both.

• Square — when one face is parallel to the projection plane.
• Rectangle — when the cube is tilted about one edge.
• Regular hexagon — the special isometric case, when the cube is balanced on a single corner vertex.
• Irregular hexagon / pentagon — for various in-between tilted orientations.

When a cube is balanced perfectly on one corner, the line from that bottom corner straight up to the diagonally opposite top corner (the cube’s space-diagonal) is exactly vertical. By the cube’s perfect symmetry around this diagonal axis (a 3-fold rotational symmetry — rotating 120° about this axis maps the cube to itself), every one of the 3 edges meeting at the bottom corner — and every one of the 3 edges meeting at the top corner — is related to the others by this rotation. Since rotation preserves lengths and angles, all edges project down to segments of exactly equal length, spaced at exactly 120° from each other.

Because of two combined geometric facts:

1. Parallel lines project to parallel lines. So every edge of the solid running along the height, length, or depth axis projects to a line in one of exactly 3 fixed directions on paper (\(|\), \(/\), \(\backslash\)) — there’s no ambiguity about which direction represents which axis.
2. The isometric projection scales all 3 axes equally. Since the cube is balanced symmetrically, a unit distance along the height axis projects to the same length as a unit distance along the length or depth axis. This means we can use a single consistent grid spacing to measure real distances along any of the 3 axes directly off the page.

In 2 dimensions, exactly 5 distinct tetromino shapes exist (I, O/square, L, S/Z, T) — these are the 5 shown in the chapter. But once we allow gluing in 3 dimensions (cubes, not flat squares), one genuinely new shape becomes possible that has no flat 2D equivalent: the 3D “twisted” tetromino, where the 4th cube is glued not in the same plane as the first 3, but on a face perpendicular to that plane — creating an L-shape that bends “out of the page.”

Build each shape edge-by-edge: for every new edge, decide which of the 3 axes (height \(|\), length \(\backslash\), depth \(/\)) it runs along, and whether it goes “up” or “down” along that direction, then draw the corresponding segment on the triangular grid. Working one cube at a time and keeping faint construction lines until the full shape is in place makes this much easier.