Scarecrow Wanders the Deadly Poppy Field

1900 Edition Illustration by W.W. Denslow

Introduction

You will recall how the travelers came to the field of poppies outside of Emerald City:

They walked along listening to the singing of the brightly colored birds and looking at the lovely flowers which now became so thick that the ground was carpeted with them. There were big yellow and white and blue and purple blossoms, besides great clusters of scarlet poppies, which were so brilliant in color they almost dazzled Dorothy’s eyes.

“Aren’t they beautiful?” the girl asked, as she breathed in the spicy scent of the bright flowers.

“I suppose so,” answered the Scarecrow. “When I have brains, I shall probably like them better.”

“If I only had a heart, I should love them,” added the Tin Woodman.

“I always did like flowers,” said the Lion. “They seem so helpless and frail. But there are none in the forest so bright as these.”

They now came upon more and more of the big scarlet poppies, and fewer and fewer of the other flowers; and soon they found themselves in the midst of a great meadow of poppies. Now it is well known that when there are many of these flowers together their odor is so powerful that anyone who breathes it falls asleep, and if the sleeper is not carried away from the scent of the flowers, he sleeps on and on forever. But Dorothy did not know this, nor could she get away from the bright red flowers that were everywhere about; so presently her eyes grew heavy and she felt she must sit down to rest and to sleep.

But the Tin Woodman would not let her do this.

“We must hurry and get back to the road of yellow brick before dark,” he said; and the Scarecrow agreed with him. So they kept walking until Dorothy could stand no longer. Her eyes closed in spite of herself and she forgot where she was and fell among the poppies, fast asleep.

“What shall we do?” asked the Tin Woodman.

“If we leave her here she will die,” said the Lion. “The smell of the flowers is killing us all. I myself can scarcely keep my eyes open, and the dog is asleep already.”

It was true; Toto had fallen down beside his little mistress. But the Scarecrow and the Tin Woodman, not being made of flesh, were not troubled by the scent of the flowers.

“Run fast,” said the Scarecrow to the Lion, “and get out of this deadly flower bed as soon as you can. We will bring the little girl with us, but if you should fall asleep you are too big to be carried.”
So the Lion aroused himself and bounded forward as fast as he could go. In a moment he was out of sight.

Scarecrow and Tin Man are left in the field of poppies, standing over the sleeping Dorothy. Toto has curled up just two yards East and two yards North of Dorothy and fallen asleep there.

“Think”, Scarecrow mutters to himself, “think! There must be something we can do! Oh, if only I had some brains!”

Desperately clutching at his straw head, the poor Scarecrow wanders fretfully among the poppies. He walks about in a random fashion: with each step he travels one yard, but the step is equally likely to be either East, West, North or South. (Due to a peculiar compulsive tenency he has regarding travel through flowery meadows, each step has to be in one of the cardinal directions.) The Tin Man remains by Dorothy’s side, holding the hand of the sleeping girl.

Scarecrow takes many, many random steps. It is possible that he may, from time, to time, step on Toto, but since Scarecrow is made of straw this will not hurt the dear pooch in the least—in fact, it would not even wake him up.

Suddenly Scarecrow has an idea: he and the Tin Man can carry Dorothy out of the field, making a sort of chair with their four hands.

Looking up, Scarecrow sees the Tin Man in the distance keeping his watch over Dorothy. Eager to put his plan into action, the Scarecrow heads back. Each of his steps is still one yard in one of the four directions (E, W, N or S) but now he takes care never to step away from the Dorothy and the Tin Man, only toward them. He also resolves not to step on Toto along the way.

Questions

How many steps does Scarecrow need to get back to Dorothy?
How many times does Scarecrow step on Toto?
How many places does Scarecrow revisit in the course of his random wandering? (Include the Dorothy’s resting place, where he started, as a possibility. Of course you should not count his walk back as part of the wandering.)

Example

Suppose that Scarecrow were to take only the following ten steps:

E
E
W
N
E
N
E
S
E
N

After his first step, he is 1 yard East of Dorothy.
After his second step, he is 2 yards East of Dorothy.
His third step takes him back a yard, so now he is 1 yard East of Dorothy. (He was previously here after Step One.)
After his fourth step, he is 1 yard East and 1 yard North of Dorothy.
After his fifth step, he is 2 yards East and 1 yard North of Dorothy.
After his sixth step, he is 2 yards East and 2 yards North of Dorothy. (He steps on Toto!)
After his seventh step, he is 3 yards East and 2 yards North of Dorothy.
After his eighth step, he is 3 yards East and 1 yard North of Dorothy.
After his ninth step, he is 4 yards East and 1 yard North of Dorothy.
After his tenth step he is 4 yards East and 2 yards North of Dorothy.

So during random journey, Scarecrow stepped on Toto once, and revisited one place. By the end of his tenth step he is four yards East and two yards North of Dorothy, so the walk back requires

\[4 + 2 = 6\]

steps.

In this example, the answers to the questions are:

The Data

Here is the input data, giving the directions of each of Scarecrow’s random steps.