2. In AO-language, all words consist only of A's and O's, and every possible sequence of A's and O's is a word. For example, there are eight three-letter words: 'OOO', 'OOA', 'OAO',..., 'AAO' and 'AAA'. Words that contain the letter combination 'AO' as often as the letter combination 'OA' are special. For instance, 'AOAAOOOAA' is special because the word has the combination 'AO' twice and also the combination 'OA' twice.

Give a special word consisting of four A's and four O's, such that if you remove any single letter, the remaining seven-letter word is also special.
B2. 'AAOOOOAA' (of 'OOAAAAOO') A word in the AO language consists of alternating blocks of A's and blocks of O's. For example, 'AA O AAA OOO AA O' has three A-blocks and three O-blocks. The letter combinations 'AO' correspond precisely to the transitions from an A-block to an O-block, and the letter combinations 'OA' correspond to the transitions from an O-block to an A-block. The word above thus has three 'AO' combinations and two 'OA' combinations. A word that starts and ends with the same letter has an equal number of transitions from A to O and vice versa, and thus an equal number of 'AO' and 'OA' letter combinations. A word that starts with A and ends with O has one more 'AO' combination, while a word that starts with O and ends with A has exactly one more 'OA' combination.

A word in the AO language is thus special precisely when the first and last letters of the word are the same. We are therefore looking for a word with the property that the first and last letters are the same. Moreover, this property must remain true when you remove the first or last letter. This means that the first two letters of the word and the last two letters of the word must all be the same. The only two words with four A's and four O's that meet this criterion are AAOOOOAA and OOAAAAOO.