Let us now consider another form of Table which might readily occur to a boy playing with his marbles, or to a young lady with the balls of her solitaire board. {51}
The boy may place a row of his marbles on the sand, at equal distances from each other, thus—
He might then, beginning with the second, place two other marbles under each, thus—
He might then, beginning with the third, place three other marbles under each group, and so on; commencing always one group later, and making the addition one marble more each time. The several groups would stand thus arranged—
He will not fail to observe that he has thus formed a series of triangular groups, every group having an equal number of marbles in each of its three sides. Also that the side of each successive group contains one more marble than that of its preceding group.
Now an inquisitive boy would naturally count the numbers in each group and he would find them thus—
1 3 6 10 15 21