Tipping Angle Game

About Topple

Topple shows a block standing on a rough ramp. The width of its base is given, and a white dot marks its center of mass, with the height of that dot above the ground written next to it. The ramp is tilted slowly until the block tips over, and you estimate at which angle, in degrees, that happens.

The ramp is rough enough that the block never slides, so the only way it can go is over. That removes friction from the question and leaves pure geometry. On the first levels the block is uniform and its center of mass sits in the middle, so tall narrow blocks tip early and wide flat blocks late. From level 5 the center of mass can be anywhere inside the block, as if a heavy load were packed high or low.

How to play

1. Look at the base width and the height of the dot. A wide base and a low dot make a stable block.
2. Imagine tilting the whole picture. The block tips when the dot passes directly above the lowest corner of the base.
3. Slide to the angle in degrees where you expect that to happen, anywhere from 1 to 89.
4. Lock in. The scene tilts to the true angle so that you can see the dot hanging over the corner.

A block stands on a rough ramp that slowly tilts. The dot marks its centre of mass. Slide to the angle, in degrees, where it tips over.

How scoring works

The score is based on how many degrees you are off. You get 100 for the exact angle and 0 when you are 24 degrees away on level 1, or 12 degrees away on level 10. In between the score falls in a straight line, so an error of 6 degrees on the top level still earns about 50 points.

The true angle is the arc tangent of half the base width divided by the height of the center of mass. It is rounded to a whole degree, and the answer timer starts at 16 seconds and shrinks to 12 seconds.

What it trains

This game trains your sense of stability: how the shape and weight distribution of an object decide how far it can be tilted. The same intuition tells you whether a stack of boxes will survive a bumpy ride, whether a tall bookcase needs to be fixed to the wall, or how far you can lean a ladder.

It is also a feel for angles of slopes. Many people underestimate how steep a ramp must be before a stable-looking block goes over, and the game quickly shows how far off they are.

Tips for a higher score

• Use the 45-degree anchor. A uniform square block tips at exactly 45 degrees, because its dot sits as high as it is wide at the half-width. A block twice as tall as wide tips at about 27 degrees.
• Compare the half-width with the height of the dot. If the dot is as high as half the width, the tipping angle is 45 degrees; if it is twice as high, about 27; if only half as high, about 63.
• Move the dot in your mind. Lowering the center of mass raises the tipping angle, which is why loaded trucks stay stable when the heavy cargo is at the bottom.
• Remember that only the position of the dot matters, not the weight of the block. A heavy and a light block with the same shape and dot tip at the same angle.
• Use the reveal animation to learn: watch how the dot ends up straight above the corner.

The idea behind it

An object tips when the vertical line through its center of mass falls outside its base of support. On an incline, the block tips about its lower edge when tan(angle) equals half the base width divided by the height of the center of mass. For a uniform block this is simply the width divided by the height.

Vehicle designers use this idea for rollover ratings. The static stability factor, which is half the track width divided by the height of the center of gravity, is used by road safety agencies to rate how likely an SUV is to roll over. It is the same ratio as in the game.

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