Can someone explain these strange properties of $10, 11, 12$ and $13$?

So my 7 year old son pointed out to me something neat about the number 12: if you multiply it by itself, the result is the same as if you took 12 backwards multiplied by itself, then flipped the result backwards. In other words: $$12 × 12 = 144$$$$21 × 21 = 441$$

I confidently explained to him that this was merely a coincidence.

But he then casually pointed out that the same holds true for 10, 11 and 13 (as long as you use leading zeros), $$10 × 10 = 100$$$$01 × 01 = 001$$

As if this wasn't enough, he also went on to mention that the same holds true for addition for those same four numbers! $$12 + 12 = 24$$ $$21+21 = 42$$

So needless to say, this is hard to chalk up to mere coincidence. Is there some non-coincidental reason for these strange findings?

Solution 1:

Suppose we have a 2 digit number $x$. We can write is in terms of its digits $a$ and $b$. When we attempt to square this number, we get an interesting result.

$$x^2=(10a+b)^2=100a^2+10(2ab)+b^2$$

We can also flip the digits (I'll use $\bar x$ to indicate this) and then square.

$$\bar x^2=(10b+a)^2=100b^2+10(2ab)+a^2$$

This result isn't very useful on its own, but if $a^2$, $b^2$, and $2ab$ are all less than $10$, then the three terms above are the three digits of $x^2$ and $\bar x^2$ respectively. It is clear from that that reversing the digits of $x$ reverses the digits of $x^2$ provided it meets those requirements. (If we switch $a$ and $b$, the first and last terms switch while the middle term is unchanged.)

Note that $10$, $11$, $12$, and $13$ (as well as $20$, $21$, $22$, $30$, and $31$) all satisfy the same condition $a^2,b^2,2ab<10$ and thus have the property you describe.

The same argument can be used for addition, since, if no digit is greater than $4$, we can add the digits individually. Rearranging the digits of such a number will apply the same rearrangement to its sum with itself.

We can play the same game with three digit numbers, but the restrictions are even greater:

$$x^2=(100a+10b+c)^2=10000a^2+1000(2ab)+100(b^2+2ac)+10(2bc)+c^2$$

If each factor multiplying a power of ten is less than ten, we have the same property. This gives us a few numbers, which you can verify all have the property.

$$100,101,102,103,110,111,112,113,120,121,122,130,200,201,202,210,211,212,220,300,301,310,311$$

Note that we have not shown that these conditions give you all the numbers for which $\bar x^2=\bar{x^2}$, though I haven't been able to find a counterexample. This argument nonetheless applies to all of the numbers you provided.

Solution 2:

Let's start with addition. In a positional system like the one we ordinarily use, when we add two numbers, digits of different weights interact through carries. If there are no carries out of any positions, the sum may be performed on each pair of digits independently and then the overall result assembled from those partial results by just arranging them in the right order. For $0 \leq i < 5$, we have $2i < 10$; hence if all digits of a number are less than $5$ the addition trick will work.

The multiplication trick is similar. For instance, $$112 \cdot 112 = 11200 + 01120 + 00224 = 12544 \enspace,$$ while $$211 \cdot 211 = 42200 + 02110 + 00211 = 44521 \enspace.$$ Note that there are no carries generated, either while computing the partial products or while computing the final sum.

While the condition $0 \leq i < 5$ works for any numbers of digits when computing $n+n$, the increase in the number of partial products means that the multiplication trick only works for relatively small numbers. For instance,

$$ 11111112 \cdot 11111112 = 123456809876544 \enspace,$$

but

$$ 21111111 \cdot 21111111 = 445679007654321 \enspace. $$

The effect of the carries is initially visible in the middle digits, which depend on the most partial products.