Question

Prove that:
$open vertical bar table row cell straight a plus straight b end cell cell space space straight b plus straight c end cell cell space space space straight c plus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell row cell straight c plus straight a end cell cell space space straight a plus straight b end cell cell space space space straight b plus straight c end cell end table close vertical bar space equals space 2 space open vertical bar table row straight a cell space space straight b end cell cell space straight c end cell row straight b cell space space straight c end cell cell space straight a end cell row straight c cell space straight a end cell cell space straight b end cell end table close vertical bar.$

Solution

Let $increment space equals space open vertical bar table row cell straight a plus straight b end cell cell space space space straight b plus straight c end cell cell space space straight c plus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space straight a plus straight b end cell row cell straight c plus straight a end cell cell space space straight a plus straight b end cell cell space space space straight b plus straight c end cell end table close vertical bar$
$equals space open vertical bar table row straight a cell space space straight b plus straight c end cell cell space space straight c plus straight a end cell row straight b cell space space straight c plus straight a end cell cell space straight a plus straight b end cell row straight c cell space space straight a plus straight b end cell cell space space straight b plus straight c end cell end table close vertical bar plus space open vertical bar table row straight b cell space space straight b plus straight c end cell cell space straight c plus straight a end cell row straight c cell space straight c plus straight a end cell cell space straight a plus straight b end cell row straight a cell space straight a plus straight b end cell cell space straight b plus straight c end cell end table close vertical bar equals space open vertical bar table row straight a cell space space space straight b end cell cell space space space straight c plus straight a end cell row straight b cell space space space straight c end cell cell space space space straight a plus straight b end cell row straight c cell space space straight a end cell cell space space space straight b plus straight c end cell end table close vertical bar space plus space open vertical bar table row straight a cell space space space straight c end cell cell space space space straight c plus straight a end cell row straight b cell space space space space straight a end cell cell space space space straight a plus straight b end cell row straight c cell space space space space straight b end cell cell space space space space straight b plus straight c end cell end table close vertical bar plus space open vertical bar table row cell straight b space end cell cell space space straight b end cell cell space space straight c plus straight a end cell row straight c cell space space space straight c end cell cell space space straight a plus straight b end cell row straight a cell space space straight a end cell cell space space space straight b plus straight c end cell end table close vertical bar plus space open vertical bar table row straight b cell space space space straight c end cell cell space space straight c plus straight a end cell row straight c cell space space space straight a end cell cell space space straight a plus straight b end cell row straight a cell space space space straight b end cell cell space space space straight b plus straight c end cell end table close vertical bar equals space open vertical bar table row straight a straight b straight c row straight b straight c straight a row straight c straight a straight b end table close vertical bar space plus space open vertical bar table row straight a straight b straight a row straight b straight c straight b row straight c straight a straight c end table close vertical bar space plus space open vertical bar table row straight a straight c straight c row straight b straight a straight a row straight c straight b straight b end table close vertical bar space plus space open vertical bar table row straight a straight c straight a row straight b straight a straight b row straight c straight b straight c end table close vertical bar plus 0 plus open vertical bar table row straight b straight c straight c row straight c straight a straight a row straight a straight b straight b end table close vertical bar plus open vertical bar table row straight b straight c straight a row straight c straight a straight b row straight a straight b straight c end table close vertical bar equals space open vertical bar table row straight a straight b straight c row straight b straight c straight a row straight c straight a straight b end table close vertical bar plus 0 plus 0 plus 0 plus 0 plus 0 plus open vertical bar table row straight b straight c straight a row straight c straight a straight b row straight a straight b straight c end table close vertical bar equals space open vertical bar table row straight a straight b straight c row straight b straight c straight a row straight c straight a straight b end table close vertical bar plus open vertical bar table row straight a straight b straight c row straight b straight c straight a row straight c straight a straight b end table close vertical bar equals 2 space open vertical bar table row straight a straight b straight c row straight b straight c straight a row straight c straight a straight b end table close vertical bar.$

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