The left and right shoes are differentiable.let the shoes be

L1,L2,L3 ; R1,R2,R3 ;

where L and R represent left and right.

It is obvious that $L1+L2+L3=R1+R2+R3$

STEP 1:

[ L1,L2,R1 ] | [ L3,R2,R3 ]

Conclusions:-

• If the weights are equal then L3 pairs with R1 and after this,only one more weighing is sufficient.
• If they are unequal then continue with next step.

STEP 2:

[ L1,L2,R2 ] | [ R1,L3,R3 ]

Conclusions:-

• If the weights are equal then L3 pairs with R2.
• If the weights are unequal then L3 pairs with R3.

STEP 3:

[ L1,R1] | [ L2,R2 ]

Conclusions:

• If the weights are equal then L2 pairs with R1.
• If the weights are unequal then L2 pairs with R2.

Note that even if the left and right shoes are not differentiable,similar steps are applicable and answer remains the same.

The equation M+M = S + L is useless. It is just a trick to make the problem more complicated. And... it succeeded on tricking me haha.

Get any left shoes, and compare it with any right shoe. Maximum of two (2) trials is needed to determine it's matching right shoe.

Remember that if in two trials there is no balance observed, then the last right shoe is the matching shoe.

Then there will be two (2) unknown pairs. Maximum of one (1) trial is needed to determine it's pair shoe.

$1 + 2 = \boxed{ 3 }$ Tada! :D

Given, L,M,H:

let take it as , light pair L= l+l medium pair P= p+p heavy pair H= h+h

There are 9 possibilities of weigh the shoes are ,

( l , m ); ( l , h ); ( l , l ); ( m , l ); ( m , m ); ( m , h ); ( h , l ); ( h , m ); ( h , h );

Here, (l,m);(m,l) : (m,h);(h,m) : (l,h);(h,l) these 6 possibilities are relatively same weight with mismatched pair.

(l,l) : (m,m) : (h,h) these are unique weight and with perfect pair;

The number of weighing required to separate all the pairs ,

Minimum number of weighing : 6 --> This is the answer

maximum number of weighing : 9

given , L+H = M+M

From above we can derive the equation to get for each pair weight,

-> L + H = M + M

-> L + H = M(1 + 1)

-> L + H = 2M

-> $M=\frac { L+H }{ 2 }$

You can prove the given equation by giving any values to L,M,H respectively.