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Part of 2003 Turkey Team Selection Test

Problems(1)

M = {(a,b,c,d)|a,b,c,d € {1,2,3,4} and abcd > 1}

Source: Turkey TST 2003 - P1

4/6/2013
Let M={(a,b,c,d)∣a,b,c,d∈{1,2,3,4} and abcd>1}M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}. For each n∈{1,2,…,254}n\in \{1,2,\dots, 254\}, the sequence (a1,b1,c1,d1)(a_1, b_1, c_1, d_1), (a2,b2,c2,d2)(a_2, b_2, c_2, d_2), …\dots, (a255,b255,c255,d255)(a_{255}, b_{255},c_{255},d_{255}) contains each element of MM exactly once and the equality ∣an+1−an∣+∣bn+1−bn∣+∣cn+1−cn∣+∣dn+1−dn∣=1|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_n|+|d_{n+1} - d_n| = 1 holds. If c1=d1=1c_1 = d_1 = 1, find all possible values of the pair (a1,b1)(a_1,b_1).
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