magic with 4 tools in positions switching

Question

There are 4 tools, on a table, in a row:

spoon,tablespoon,fork,knife

in positions:

4,3,2,1 respectively.

The magician look at their initial arrangement and then turns around. Then a volunteer from the crowd comes by and starts switching between pairs of tools over and over again.

First he makes 10 switches and declares the positions of the tools he switched then he makes one "silent" switch (without saying anything). He then makes 10 more "loud" switches.

After this process the magician turns to the table, looks at the row of tools and picks up one of the tools that the volunteer switched in the "silent" switch. The magician performs the magic by only remembering one number in the range of 1 to 10 that can be changed again and again.

How does he perform the magic?

Answer 1

The final permutation done by the volunteer is σ

α = σ₁₀ σ₉ σ₈ σ₇ σ₆ σ₅ σ₄ σ₃ σ₂ σ₁
β = σ₂₀ σ₁₉ σ₁₈ σ₁₇ σ₁₆ σ₁₅ σ₁₄ σ₁₃ σ₁₂ σ₁₁
σ = β σₛ α

where σᵢ are transpositions and σₛ is the silent transposition.

Then, the magician can

choose two different numbers a and b in {1,2,3,4}.

Each time the volunteer makes a "loud" transposition, the magician

does it too on his numbers.

Finally, he gets

β α (a) and β α (b)

Then, he checks

whether the objects which were initially in positions a and b are now in positions β α (a) and β α (b).

• If both are, that means the silent transposition didn't affect these objects. Since there are only 4 objects, the transposition affected the other two.
• If (at least) one isn't, that means it was affected by the silent transposition.

In order to achieve this, the magician must keep track of two different numbers between 1 and 4. That's 12 possibilities, which is too much for his limited memory.

However, there is a trick (I'm not sure if this is cheating, but otherwise I think it's impossible):

he does not need to use his memory in order to keep track of a number. For example, he can store the numbers in his own eyes and tongue, using a code like 1 = up, 2 = right, 3 = down, 4 = left.

When the volunteer says a loud transposition, the magician does this

1. He retrieves the number stored in his eyes and places it in his memory
2. He transforms that number according to the transposition
3. He stores the new number in his eyes
4. He retrieves the number stored in his tongue and places it in his memory
5. He transforms that number according to the transposition
6. He stores the new number in his tongue

Only one number between 1 and 4 is stored in the memory at a time.

Of course, this assumes the transposition and the conversion code do not waste memory. Otherwise, he could use his fingers to store more data.

This will work because the magician is turned around, otherwise the public would realize there is something wrong with his face: