This one can be shown to any student that understands the distance formula, and has a willingness to think about $n$-dimensional space for $n>3$.
(It is, more specifically, an indication that "inducting on pictures" is invalid, and less so of a "pay close attention to the picture". But I think it's striking enough to warrant inclusion here.)
Draw four circles with $r=1$ on the plane at the points $(\pm 1,\pm 1)$. Draw the box which perfectly encloses these circles, i.e. a square with side length 2. Now, draw a circle that is tangent to all four of the circles. Notice that this circle is inside the box.
You can repeat this for $n=3$. Draw spheres with $r=1$ at the eight points $(\pm 1,\pm 1,\pm 1)$. Draw the box that encloses these, i.e. a cube of side length 2. Now, draw a sphere that is tangent to all those spheres. Again, you can see it sits inside the box.
Repeat this in higher dimensions. When you get to $n\geq 10$, the "inner" sphere leaks outside the box!