domino spots - evincing a sprightlier attitude

how many spots ('pips') are there altogether on a 1 to 6 set of dominoes?

one neat aspect of this problem is the variety of approaches that could be adopted


















FC Boon elaborates his use of this problem in 'puzzle papers in arithmetic' (1937)
he argues for an inclusion of solving such problems in a maths curriculum:
"Matter and methods outside, and even beyond, the scope of the syllabus stimulate interest and widen ideas and pupils evince a sprightlier attitude which quickens their pace in more formal work."

with this arrangement, can you find the sums:

(a) 3(0 + 1 + 3 + 6 + 10 + 15 + 21) by using columns? 
     why does each column sum to 3 times a triangular number? 

(b) 6(7 + 6 + 5 + 4 + 3 + 2 + 1) from the diagonals?

(c) 8(0 + 1 + 2 + 3 + 4 + 5 + 6) by looking at the occurrences of each number?

generalising
to dominoes with spots from 1 to 'n'
can lead to a rule or rules for summing the triangular numbers