Instant Insanity: The Four Cubes Problem
Quick answer: Instant Insanity is four cubes, each with its six faces painted in four colours, which must be stacked in a column so that all four colours appear on each of the four sides. There are 244 = 331,776 ways to orient four cubes and essentially one that works. The reliable method is not trial and error, it is a small piece of graph theory.
Best for: maths teachers, and anyone who has spent an hour restacking four cubes and getting three sides right.
Three sides right is the trap. It happens constantly, it feels like progress, and it is almost never one twist away from a solution.
What Is Instant Insanity?
Four cubes. Each face is painted one of four colours, conventionally red, blue, green and white, and the colours are distributed differently on each cube.
Stack them into a tower one cube wide and four tall. The requirement is that looking at the front of the tower you see all four colours, and the same for the back, the left and the right. The top and bottom faces are ignored entirely.
That last detail matters more than it sounds, and it is the opening for the method below.
Where Did It Come From?
It is much older than the name, which is a 1960s marketing invention.
The puzzle goes back to around 1900. The puzzle historian Jerry Slocum traces it to Frederick A. Schossow, who marketed a version as the Katzenjammer Puzzle. It also circulated as The Great Tantalizer, which is the name most pre-war references use.
The version everyone knows was devised by Frank Armbruster and has been sold by Parker Brothers as Instant Insanity since 1967. The name was good enough that it displaced sixty years of earlier ones.
Why Does Trial and Error Fail?
Because the search space is large and the puzzle gives almost no feedback on the way.
A cube has 24 distinct orientations. Four cubes gives 244, or 331,776 combinations, and rotating the finished tower cuts that down only by a small factor. Against that, there is essentially a single correct answer.
Worse, the puzzle is not warmer or colder. A stack with three perfect sides and one repeated colour is not close to the solution in any useful sense. It can require re-orienting every cube to fix. There is no gradient to follow, which is exactly the property that makes hill-climbing useless and makes people stack the same wrong towers repeatedly.
How Do You Solve Instant Insanity With Graph Theory?
This is the real method, it is the reason the puzzle appears in textbooks, and it takes about ten minutes with a pencil.
Step 1. Draw four dots. One for each colour: R, B, G, W. This is your graph, and you will draw on the same four dots four times over.
Step 2. Turn each cube into three edges. Here is the key idea. A cube has three pairs of opposite faces. For each pair, draw an edge joining the two colours on it. If a pair has the same colour on both faces, draw a loop at that dot. Every cube becomes exactly three edges. Label each edge with which cube it came from, 1 to 4.
Why opposite faces? Because in the finished tower, the two faces of any opposite pair point in opposite directions. One contributes to the front, the other to the back. An edge is precisely a record of which side pairing that cube can supply.
Step 3. Combine all four cubes into one graph. You now have twelve edges on four dots, each tagged with its cube number.
Step 4. Find two subgraphs. This is the whole puzzle. You are looking for two edge-disjoint subgraphs, each of which:
- uses exactly one edge from each cube, so four edges in total;
- has every vertex of degree two, meaning all four colours are touched exactly twice;
- shares no edge with the other subgraph.
One subgraph will describe the front-and-back pairing. The other describes left-and-right. Degree two at every vertex is the formal way of saying each colour appears once on the front and once on the back.
Step 5. Build the tower. Read each cube's edge from the first subgraph and set that pair of colours facing front and back. Then rotate the cube about that axis until its edge from the second subgraph points left and right. The tower is solved.
The reason this works is that the top and bottom faces never mattered. Once you know the front-back pair and the left-right pair for each cube, the remaining pair is forced, and nobody is looking at it.
How Do You Solve It Without the Maths?
You can, with discipline, if you drop the habit of building towers.
Work with two sides at a time. Pick front and back only. Find an arrangement where all four colours appear front and all four appear back, ignoring the other two sides completely. Note it down.
Then test whether it survives. Spin each cube about its front-back axis and see whether left and right can also be made to work. If not, discard that front-back arrangement entirely and find a different one.
That is the graph method performed badly, and it is still vastly better than stacking and hoping, because you are discarding whole families of arrangements rather than one tower at a time.
What Should You Try Instead?
Instant Insanity is a colour-matching puzzle, and we make nothing that works that way. Our puzzles are solved by shape, not by hue, which is a real difference rather than a modest one.
If what appealed was a small number of pieces hiding one correct answer, that is Identical Twins, which is two identical pieces and a frame, and EcstaTIC, which is two pieces and a single rotation. If it was the combinatorics, Cube Alchemy is a seven-piece cube with 180 curated challenges, and the Soma cube covers that ground properly.
The wider range is in cube puzzles and mechanical puzzles.
Frequently Asked Questions
What is Instant Insanity?
Four cubes whose faces are painted in four colours, which must be stacked in a column so that each of the four vertical sides shows all four colours. The top and bottom faces do not count.
What is the solution to Instant Insanity?
There is essentially one correct stack, and it is found with graph theory rather than guessing. Represent each cube as three edges on four colour-vertices, one edge per pair of opposite faces, then find two edge-disjoint subgraphs that each use one edge per cube and touch every colour exactly twice. One gives the front-back orientation, the other left-right.
How many combinations does Instant Insanity have?
A cube has 24 orientations, so four cubes give 24 to the fourth power, which is 331,776. Rotating the whole finished tower reduces the count a little, but there is still essentially a single solution to find.
Why is Instant Insanity so hard?
Because it gives no feedback. A stack with three correct sides is not nearly solved and may need every cube re-oriented. With no way to tell whether you are getting warmer, trial and error has no direction to follow.
Who invented Instant Insanity?
The modern version was devised by Frank Armbruster and sold by Parker Brothers from 1967. The puzzle itself dates to around 1900, when Jerry Slocum traces it to Frederick A. Schossow's Katzenjammer Puzzle; it was also sold as The Great Tantalizer.
Why does the graph theory method use opposite faces?
Because in the finished tower the two faces of an opposite pair point in opposite directions, one to the front and one to the back. An edge between two colours records a side pairing that cube is able to supply, which is exactly the information the puzzle needs.
