## Two Heads are Better than One

Raquel and Esperanza were asked to count Dr. Dolittle’s ostrichs and pushimi-pullyus. Raquel counted 67 heads, while Esperana counted 134 legs.

Raquel and Esperanza were asked to count Dr. Dolittle’s ostrichs and pushimi-pullyus. Raquel counted 67 heads, while Esperana counted 134 legs.

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Cut the Pushmi-Pullyus in half so that we have 3 kinds of creatures:

Pushmis with 1 head, 2 legs.

Pullyus with 1 head, 2 legs.

Ostriches with 1 head, 2 legs.

Whatever creature we pick, each 1 head will correspond to 2 legs.

In particular, the desired total of 67 heads will correspond to 67×2 = 134 legs (also as desired!).

As long as the total is 67 heads, the modified problem achieves the right leg-count, too. In the original problem, the Pushmis and Pullyus were not cut apart; so we will need the same number of each of them (so that we can sew them back together). Call this number X. If we then call the number of ostriches Y, we end up with: X+X+Y = 67.

Re-arranged, Y = 67 – 2X.

For any X between 1 and 33 (why these numbers?), we get a corresponding Y. And so there are 33 total solutions.

A single example:

If X is 5, then 2X is 10, and we have: Y = 67 – 10 = 57.

After sewing, we have 5 Pushmi-Pullyus (10 heads, 20 legs) and 57 ostriches (57 heads, 114 legs). Adding up the parenthetical counts, we have 67 heads and 134 legs as desired.

MQ