*Notes. Act 1. Act 3.*

**ACT 2 -- THE DREAM OF THE PRACTICE EXAM** *(Music. A dream song, like Schumann's "Träumerei")***AOKI (A SENIOR):** H'm, I don't recall seeing this place before. What am I doing walking around a field like this now? Hey! That's strange. A billboard suddenly appeared before my eyes. What's that written on it? "Practice Examination Area". Well, if I pass a practice exam that might give me some extra confidence, so I wouldn't mind giving that a try... but where is this "area"?

**FEMALE TEACHER:** Excuse me, Mr Aoki? This way, please.

**AOKI:** Oookay... When did a teacher get here? Ah, there's a desk and a chair. Ah, I see. This must be the Practice Examination Area.

**FEMALE TEACHER:** All right, everyone, now that you're all here I shall begin giving you the problems. Are you ready? Here we go: "There was once an old lady who sold eggs. One day, she put a small number of eggs in a basket and set off for town to sell them. At the first house, she sold an amount equal to half of her eggs plus half of one egg. At the next house, she sold half of the rest of the eggs plus another half an egg. And at the last house, she sold half of the remaining eggs, plus another half-egg -- and then she had sold them all." How many eggs were there in the basket to begin with? I should add that she did not actually break any eggs in half -- she sold only whole eggs at all three houses. Please solve this problem using algebra.

**AOKI:** Sensei, please repeat the question.

**FEMALE TEACHER:** Yes, I'll repeat it once more. "An old lady egg-seller put a small number of eggs into a basket and set off for town to sell them. At the first house she sold half of the eggs plus half of a single egg; at the next house she sold half of the rest of the eggs plus half of an egg; and at the last house she sold half of the remaining eggs plus another half of an egg, and with that all of the eggs had been sold." How many eggs were in her basket when she set off? None of the eggs were sold broken. Please solve this problem using algebra.

**AOKI:** Gosh, how interesting. It's just like a riddle. And being told to solve it with algebra makes it even better. All right, first of all let's call the total number of eggs

*x*, and then...

**FUSAKO (A FEMALE SENIOR):** Sensei, I've finished.

**AOKI:** What the-- there's already someone finished!

**FEMALE TEACHER:** Thank you Fusako, very good. What kind of equations did you use? Please read out your work to us.

**FUSAKO:** Yes, sensei. At the beginning, the number of eggs sold at the first house is

*x* divided by two plus half of one egg. If we put this together, we have a figure whose denominator is two and numerator is

*x* plus one. We can call this

*a*.

**FEMALE TEACHER:** Call it

*a*? All right, and then...?

**FUSAKO:** Well, the number of eggs sold at the next house is half of the remaining eggs -- in other words,

*x* minus

*a* divided by two -- plus half of one egg. We have seen

*a* before, so we can substitute it here, and after that we see that the number of eggs sold at the second house is

*x* plus one, all divided by four. Let's call this

*b*.

**FEMALE TEACHER:** Excellent work so far. What happens next?

**FUSAKO:** The number of eggs sold in the last house is

*x* minus

*a* minus

*a* all divided by two, plus half of one egg. This comes out to a figure with a numerator of

*x* plus one and a denominator of eight. Call this

*c*.

**FEMALE TEACHER:** And so the answer is...?

**FUSAKO:** *x* equals

*a* plus

*b* plus

*c*, and we can substitute and solve this to find that

*x* equals seven.

**FEMALE TEACHER:** Correct! There were originally seven eggs in the basket.

△ Vigorous applause

**AOKI:** What a surprise. That girl's pretty good for a kid... I'm going to have to study algebra harder. Ahh, another teacher's appeared at the lectern, a bearded old man this time.

**OLD TEACHER:** Now see here, all of you! It appears to me that you have been utterly conquered by textbooks and swallowed by exams, and this won't do at all. You must study in a more relaxed, easy way. To this end, I shall give you a more fanciful problem. If you can solve this one properly, you'll pass the exam. This is a

solid geometry problem. Er... "Use geometry to prove the existence of spirits". All right? Do you understand? I'll explain it to you once more: "Use geometry to prove the existence of spirits." That is the entire problem.

△ Everyone begins to talk

**OLD TEACHER:** See here! Be quiet! Aoki -- give the problem a try. Come on, stand up. Stand up like a man.

**AOKI:** "Prove that spirits are real with geometry"? I can't solve a weird problem like that.

**OLD TEACHER:** It doesn't matter! I said to stand up! If you stand up, I'll gently send out an electro-magnetic wave to help you.

**AOKI:** You will? You'll help me? All right, I've stood up. Let us first consider a two-dimensional plane world. What the-- this is strange! My mouth started speaking on its own!

**OLD TEACHER:** First we consider a plane world. What do we do next? Keep going.

**AOKI:** Yes, Sensei. In a plane world, there is length and width, but no-one knows about height. We ourselves live in a three-dimensional, solid world, and so everyone understands what length, width and height are. But we can understand a plane world if we think of a still body of water. Imagine, if you please, a "world" consisting only of the surface of the water. Then please imagine the beings who would live there.

**OLD TEACHER:** Very good, very good, and then?

**AOKI:** I have one egg here. This egg is of course a solid. I hold it in my fingers and bring it close to the face of the water. Then I gently let go. Now, what happens next? Of course, the egg will pass through the surface and fall into the water. But, to the beings who live in the face-of-the-water world, what does the egg looks like as it passes through?

**OLD TEACHER:** Indeed, how does it appear, I wonder. After all, in a plane world, the egg's height cannot be comprehended.

**AOKI:** First think of the moment the egg touches the face of the water. In the face-of-the-water world, it looks like nothing but a single point. After all, they cannot see anything above or below the face of the water.

**OLD TEACHER:** H'm, I see.

**AOKI:** Next, the egg will continue to fall further and further through the water. How will this look in the face-of-the-water world? No sooner have they noticed the sudden appearance of a single point than it becomes a larger, round shape and continues to grow, even as they watch. And then before long, after it has reached its largest size, it begins to shrink again. In other words, when the egg is more than halfway into the water, its profile begins to get smaller again, and the area intersecting with the face of the water decreases. Eventually it becomes a single point again, and then vanishes altogether. In the face-of-the-water world, they don't know that an egg has fallen through. At first a point appears, then it grows larger as they watch, then begins to get smaller, then finally disappears, almost with a

*pop*. They have no idea what it was they saw. And so they think "Ah, that must have been one of those spirits I've heard about."

**OLD TEACHER:** A confused way of putting it, but not far off, I suppose. What next?

**AOKI:** Next... I will prove the reality of spirits who appear in our world of solids. We humans know nothing other than the three measurements of length, width and height, but let us suppose that right here and now, there was another world of super-solids, incorporating some measurement that humans cannot comprehend. Now, if an egg or somesuch from the world of super-solids suddenly passed before our eyes, how would it look to us? First, a single point would appear. After that, as we watched, it would swell until it was like a rubber balloon, get larger and larger until it was as large as a gas tank. Then all at once it would begin to shrink, and before we could say "look at that" it would be as small as a balloon again, and then finally vanish,

*pop*. Well, humans would be absolutely floored. What would they say? "Ah, I just saw a spirit there!"

In other words, when something from the world of super-solids comes into our world of solids, it appears to us to be a spirit.

△ Some sound effects of spirits disappearing and mysterious music.

**Source** 『家庭コント 新学期行進曲』 (

*Katei Konto Shingakki Kôshinkyoku*), broadcast from JOAK September 30, 1938, written by

Unno Jûza (

海野十三) (1897-1949)

Aozora Bunko version entered by Tsuchiya Takeshi (土屋隆) and proofread by Tanaka Tetsurô (田中哲郎)

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