Crypto Algorithm Question - Vigenere cipher

Hello,

Do you have any idea about this question:

What key which if used to encrypt the ciphertext again would give back the plaintext (key is a weak key)? Define a formula for identifying weak keys for the cipher below (assume |K1| = |K2| = |K3|)

It would be any key where every plaintext letter is shifted by a degree of half the alphabet, namely k = 'nnn...nnnn' where the period t = n. If you have a plaintext m ='agp', key k = 'nnn' and ciphertext c = 'ntc' then applying Enc_k(c) = 'agp' = c' = m because we shift all of the letters by the same degree forward to get the plaintext.