At the end of the operation the beads will be distributed as follows:

2beadsonthefirstwire(units)
5"""second"(tens)
1"""third"(hundreds)
4"""fourth"(thousands)
6"""fifth"(tens of thousands)
2"""sixth"(hundreds of thousands)

This distribution translated into figures gives the following number: 264,152. This may be written as a result right after the factors without the partial products: that is, 742 × 356 = 264,152.

Although this description may sound very complicated, the exercise on the counting-frame is an easy and most interesting arithmetic game. And this game, which contains the secret of such surprising results, not only is an exercise which makes more and more clear the decimal relations of reciprocal value and position, but also it explains the manner of procedure in abstract operations.

Fig. 1. The disposition of the beads for the number 49,152.

In fact, in the multiplication as commonly performed:

356 ×
742
712
1424
2492
264152

the same operations are involved; but the figures, once written down, cannot be modified as is possible on the frame by moving the beads and substituting beads of higher value for those of lower value when the ten beads of one wire, as a mechanical result of the structure of the frame, are all used. As multiplication is ordinarily written, such substitutions cannot be made; but the partial products must be written down in order, placed in column according to their value, and finally added. This is a much longer piece of work, because the act of writing a figure is more complicated than that of moving a bead which slides easily on the metal wire. Again, it is not so clear as the work with the beads, once the child is accustomed to handling the frame and no longer has any doubt as to the position of the different values, and when it has become a sort of routine to substitute one bead of the lower wire for the ten beads of the upper wire which have been exhausted. Furthermore, it is much easier to add new products without the possibility of making a mistake. Let us go back to the point in the operation where the beads on the frame read thus:

2
5
1
9
1