The villagers in Lambda Village were struggling with this difficult problem:
Hmm… I have no idea.
But we need to solve this to get our mathboxes back.
Looking for some other page?
That’s when Saya , a little girl who lives in Lambda Village, appeared.
Hey guys! What are you doing there?
Hi Saya. Well, we have some bad news. The devil has taken our mathboxes .
To get them back, we need to solve this difficult question. But it’s too hard for us.
I see. I’ll help you then!
We appreciate it, but this isn’t for kids.
We tried hard but couldn’t solve it. I don’t think you can do it.
Saya spent some time thinking, and came up with the following lunchbox:
Hey, I filled each icon as follows. Can you take a look?
Hmm… I don’t think it’s going to work, but we’ll try anyway.
We’ll run this, and see if the result is a lunchbox that can be converted to .
Let’s run it!
First: Because there are multiple pairs of 1’s, we start from the leftmost pair.
Press Run . (Or press to step through manually.)
Second: We can’t start with the leftmost pair of 1’s because there’s only one item on the bottom (a sandwich ).
Therefore, we do the pair of 2’s.
Press Run :
Third: We’ll need to start with the pair of 2’s again, just like the last time.
Press Run :
So this is the final result.
Oh…! This is…
The resulting lunchbox can be converted to .
So, it’s a success!
Yay!
No way! Can’t believe Saya solved it so easily!
Surprise: Saya just solved the devil’s difficult problem.
Wow, Saya, you’re awesome. Thank you for solving it!
Sure thing!
Thanks to Saya, we’ll get our mathboxes back!
Yes, but: Before we move on, let’s dive in a little deeper.
We’ll spend a bit of time to talk about some important stuff.
Hmm… ok, let me hear what you have to say.
First: Here’s what we’ve learned so far. Here’s a lunchbox that can be converted to :
If we combine it with Saya’s lunchbox, it will become a lunchbox that can be converted to .
So, by using Saya’s lunchbox,
Now, here’s a question: Suppose that we combine Saya’s lunchbox with a lunchbox that can be converted to .
What do you think happens when you run it?
Hmm… I have no idea…
Last time, became . Don’t know what will happen this time…
Let’s check!
This time, we’ll use this lunchbox that can be converted to .
Let’s combine it with Saya’s lunchbox.
Let’s run it. This time, we’ll continue running until the end without stopping.
You can also skip to the end.
The resulting lunchbox can be converted to .
That means…
That means: By using Saya’s lunchbox, became .
So, by using Saya’s lunchbox:
Do you see the pattern?
Maybe…it can add ?
Exactly!
In other words: becomes .
That’s why “it can add .”
Now, let’s recall that: Mathboxes have the “Plus 1” feature .
Using a mathbox, you can use the “Plus 1” feature to add to any number.
And on this page, we learned that: You can also add to a number using a lunchbox .
Here’s how: Instead of , by using Saya’s lunchbox, you can add to a number.
Therefore: A mathbox that uses the “Plus 1” feature can be “reproduced” using a lunchbox.
So, to add 1 to a number:
Of course, you can also reproduce more complicated mathboxes.
For example, take a look at this mathbox that appeared on Beginner 1. If you run it, the result will be .
Can you figure out how to reproduce this using a lunchbox?
I got it!
First, we’ll use this lunchbox that can be converted to :
Then, use my lunchbox twice:
Correct! Let’s run it to check.
(To save time, we’ll skip right to the end when you run it.)
The resulting lunchbox can be converted to .
Therefore, we were able to reproduce the earlier mathbox.