Question

Show that:
$open vertical bar table row cell straight b plus straight c end cell cell space straight c plus straight a end cell cell space straight a plus straight b end cell row cell straight q plus straight r end cell cell space straight r plus straight p end cell cell space straight p plus straight q end cell row cell straight y plus straight z end cell cell space straight z plus straight x end cell cell space straight x plus straight y end cell end table close vertical bar space equals space 2 open vertical bar table row straight a cell space space space space straight b end cell cell space space straight c end cell row straight p cell space space space straight q end cell cell space space straight r end cell row straight x cell space space straight y end cell cell space space straight z end cell end table close vertical bar$

Solution

Let $increment space equals space open vertical bar table row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space straight a plus straight b end cell row cell straight q plus straight r end cell cell space straight r plus straight p end cell cell space space straight p plus straight q end cell row cell straight y plus straight z end cell cell space straight z plus straight x end cell cell space space straight x plus straight y end cell end table close vertical bar$
$equals space open vertical bar table row straight b cell space space space straight c plus straight a end cell cell space space space straight a plus straight b end cell row straight q cell space space space straight r plus straight p end cell cell space space space straight p plus straight q end cell row straight y cell space space space straight z plus straight x end cell cell space space space straight x plus straight y end cell end table close vertical bar plus space open vertical bar table row straight c cell space space straight c plus straight a end cell cell space space straight a plus straight b end cell row straight r cell space space straight r plus straight p end cell cell space space straight p plus straight q end cell row straight z cell space straight z plus straight x end cell cell space space straight x plus straight y end cell end table close vertical bar equals space open vertical bar table row straight b cell space space straight c end cell cell space space straight a plus straight b end cell row straight q cell space space straight r end cell cell space space straight p plus straight q end cell row straight y cell space straight z end cell cell space space straight x plus straight y end cell end table close vertical bar plus space open vertical bar table row straight b cell space space space straight a end cell cell space space space straight a plus straight b end cell row straight q cell space space space straight p end cell cell space space space straight p plus straight q end cell row straight y cell space space space straight x end cell cell space space space space straight x plus straight y end cell end table close vertical bar plus space open vertical bar table row straight c straight c cell straight a plus straight b end cell row straight r straight r cell straight p plus straight q end cell row straight z straight z cell straight x plus straight y end cell end table close vertical bar plus open vertical bar table row straight c cell space straight a end cell cell space straight a plus straight b end cell row straight r cell space straight p end cell cell space straight p plus straight q end cell row straight z cell space straight x end cell cell space straight x plus straight y end cell end table close vertical bar equals space open vertical bar table row straight b cell space space space space straight c end cell cell space space space straight a end cell row straight q cell space space space space straight r end cell cell space space space straight p end cell row straight y cell space space straight z end cell cell space space space straight x end cell end table close vertical bar plus open vertical bar table row straight b cell space space straight c end cell cell space space straight b end cell row straight q cell space space straight r end cell cell space space straight q end cell row straight y cell space straight z end cell cell space space straight y end cell end table close vertical bar plus open vertical bar table row straight b cell space space straight a end cell cell space space straight a end cell row straight q cell space space straight p end cell cell space space straight p end cell row straight y cell space space straight x end cell cell space space straight x end cell end table close vertical bar plus space open vertical bar table row straight b cell space space space straight a end cell cell space space straight b end cell row straight q cell space space space straight p end cell cell space space straight q end cell row straight y cell space space straight x end cell cell space space straight y end cell end table close vertical bar plus 0 plus space open vertical bar table row straight c cell space space straight a end cell cell space space space straight a end cell row straight r cell space space straight p end cell cell space space space straight p end cell row straight z cell space straight x end cell cell space space space straight x end cell end table close vertical bar space plus space open vertical bar table row straight c cell space space straight a end cell cell space space space straight b end cell row straight r cell space space straight p end cell cell space space space straight q end cell row straight z cell space straight x end cell cell space space space straight y end cell end table close vertical bar equals space open vertical bar table row straight b cell space space space straight c end cell cell space space straight a end cell row straight q cell space space space straight r end cell cell space space straight p end cell row straight y cell space space straight z end cell cell space space straight x end cell end table close vertical bar plus 0 plus 0 plus 0 plus 0 plus 0 plus open vertical bar table row straight c cell space space space straight a end cell cell space space space straight b end cell row straight r cell space space space straight p end cell cell space space space straight q end cell row straight z cell space space space space straight x end cell cell space space space straight y 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 end cell row straight p cell space space space straight q end cell cell space space space straight r end cell row straight x cell space space straight y end cell cell space space space straight z end cell end table close vertical bar plus space open vertical bar table row straight a cell space space straight b end cell cell space space space straight c end cell row straight p cell space space straight q end cell cell space space space straight r end cell row straight x cell space space straight y end cell cell space space space straight z end cell end table close vertical bar space equals space 2 space open vertical bar table row straight a cell space space space straight b end cell cell space space straight c end cell row straight p cell space space straight q end cell cell space space straight r end cell row straight x cell space space straight y end cell cell space space straight z end cell end table close vertical bar$

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