Maximum possible city block distanceAnya participated in a study comparing two blood glucose monitors. A total of 20 persons with diabetes participated. Ten participants used Monitor A for two days and then used Monitor B for two days. The other ten participants (including Anya) used Monitor B for two days and then used Monitor A for two days. On the 5th day, participants rated the two monitors on the basis of four features. The four features were rated on a 5 point scale that ranged from −2 (poorly designed) to 2 (well designed). The distance between Anya’s ratings of the two monitors was 4 city blocks. What is the maximum possible city block distance between the two blood glucose monitors?

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Arturo Mays · Accepted Answer

A participant’s rating of a monitor is a 4-tuple   ⟨      r    1    ,      r    2    ,      r    3    ,      r    4    ⟩, where rk is the rating on feature k for   k  =  1  ,  2  ,  3  ,  4; each       r    k    ∈  {  −  2  ,  −  1  ,  0  ,  1  ,  2  }. If   r  =  ⟨      r    1    ,      r    2    ,      r    3    ,      r    4    ⟩and   s  =  ⟨      s    1    ,      s    2    ,      s    3    ,      s    4    ⟩ are two ratings, the city block distance between them is  d  (  r  ,  s  )  =      ∑          k      =      1        4        |        r    k    −      s    k        |      .For example, if   r  =  ⟨  −  2  ,  0  ,  2  ,  1  ⟩ and   s  =  ⟨  −  1  ,  −  1  ,  −  2  ,  2  ⟩, then                    d        (        r        ,        s        )                            =                  |                −        2        −        (        −        1        )                  |                +                  |                0        −        (        −        1        )                  |                +                  |                2        −        (        −        2        )                  |                +                  |                1        −        2                  |                                                  =                  |                −        1                  |                +                  |                1                  |                +                  |                4                  |                +                  |                −        1                  |                                                  =        7                .            The maximum possible value of       |        r    k    −      s    k        |   occurs when one of       r    k   and       s    k   is 2 and the other is −2, so the maximum possible value of d(r,s) is ... ?

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