Binary Babushkas

What’s the connection between grandmothers and this set of numbers?


1, 2, 6, 12, 44, 92, 184, 1208, 1256, 4792, 9912, 19832, 39664, 563952, 576464, 4496112, 4499184, 17996528, 17997488, 143972080, 145057520, 145070832, 294967024, 589944560...

To take the first step towards the answer, you need to put the numbers into binary:


1, 10, 110, 1100, 101100, 1011100, 10111000, 10010111000, 10011101000, 1001010111000, 10011010111000, 100110101111000, 1001101011110000, 10001001101011110000, 10001100101111010000, 10001001001101011110000, 10001001010011011110000, 1000100101001101011110000, 1000100101001111010110000, 1000100101001101011011110000, 1000101001010110011011110000, 1000101001011001101011110000, 10001100101001101011011110000, 100011001010011101011011110000...

The second step is compare those binary numbers with these binary numbers, which represent 1 to 30:


1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110...

To see what’s going on, take the first five numbers from each set:


• 1, 10, 110, 1100, 101100...
• 1, 10, 11, 100, 101...

What’s going on? If you look, you can see the n-th binary number of set 1 contains the digits of all binary numbers <= n in set 2. For example, 101100 is the 5th binary number in set 1, so it contains the digits of the binary numbers 1 to 5:


101100 ← 1
101100 ← 10
101100 ← 11
101100 ← 100
101100 ← 101

Now try 1256 = 10,011,101,000, the ninth number in set 1. It contains all the binary numbers from 1 to 1001:


10011101000 ← 1 (n=1)
10011101000 ← 10 (n=2)
10011101000 ← 11 (n=3)
10011101000 ← 100 (n=4)
10011101000 ← 101 (n=5)
10011101000 ← 110 (n=6)
10011101000 ← 111 (n=7)
10011101000 ← 1000 (n=8)
10011101000 ← 1001 (n=9)

But where do grandmothers come in? They come in via this famous toy:

Nested doll or Russian doll

It’s called a Russian doll and the way all the smaller dolls pack inside the largest doll reminds me of the way all the smaller numbers 1 to 1010 pack into 1001010111000. But in the Russian language, as you might expect, Russian dolls aren’t called Russian dolls. Instead, they’re called matryoshki (матрёшки, singular матрёшка), meaning “little matrons”. However, there’s a mistaken idea in English that in Russian they’re called babushka dolls, from Russian бабушка, babuška, meaning “grandmother”. And that’s what I thought, until I did a little research.

But the mistake is there, so I’ll call these babushka numbers or grandmother numbers:


1, 2, 6, 12, 44, 92, 184, 1208, 1256, 4792, 9912, 19832, 39664, 563952, 576464, 4496112, 4499184, 17996528, 17997488, 143972080, 145057520, 145070832, 294967024, 589944560...

They’re sequence A261467 at the Online Encyclopedia of Integer Sequences. They go on for ever, but the biggest known so far is 589,944,560 = 100,011,001,010,011,101,011,011,110,000 in binary. And here is that binary babushka with its binary babies:


100011001010011101011011110000 ← 1 (n=1)
100011001010011101011011110000 ← 10 (n=2)
100011001010011101011011110000 ← 11 (n=3)
100011001010011101011011110000 ← 100 (n=4)
100011001010011101011011110000 ← 101 (n=5)
100011001010011101011011110000 ← 110 (n=6)
100011001010011101011011110000 ← 111 (n=7)
100011001010011101011011110000 ← 1000 (n=8)
100011001010011101011011110000 ← 1001 (n=9)
100011001010011101011011110000 ← 1010 (n=10)
100011001010011101011011110000 ← 1011 (n=11)
100011001010011101011011110000 ← 1100 (n=12)
100011001010011101011011110000 ← 1101 (n=13)
100011001010011101011011110000 ← 1110 (n=14)
100011001010011101011011110000 ← 1111 (n=15)
100011001010011101011011110000 ← 10000 (n=16)
100011001010011101011011110000 ← 10001 (n=17)
100011001010011101011011110000 ← 10010 (n=18)
100011001010011101011011110000 ← 10011 (n=19)
100011001010011101011011110000 ← 10100 (n=20)
100011001010011101011011110000 ← 10101 (n=21)
100011001010011101011011110000 ← 10110 (n=22)
100011001010011101011011110000 ← 10111 (n=23)
100011001010011101011011110000 ← 11000 (n=24)
100011001010011101011011110000 ← 11001 (n=25)
100011001010011101011011110000 ← 11010 (n=26)
100011001010011101011011110000 ← 11011 (n=27)
100011001010011101011011110000 ← 11100 (n=28)
100011001010011101011011110000 ← 11101 (n=29)
100011001010011101011011110000 ← 11110 (n=30)

Babushka numbers exist in higher bases, of course. Here are the first thirteen in base 3 or ternary:


1 contains 1 (c=1) (n=1)
12 contains 1, 2 (c=2) (n=5)
102 contains 1, 2, 10 (c=3) (n=11)
1102 contains 1, 2, 10, 11 (c=4) (n=38)
10112 contains 1, 2, 10, 11, 12 (c=5) (n=95)
101120 contains 1, 2, 10, 11, 12, 20 (c=6) (n=285)
1021120 contains 1, 2, 10, 11, 12, 20, 21 (c=7) (n=933)
10211220 contains 1, 2, 10, 11, 12, 20, 21, 22 (c=8) (n=2805)
100211220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100 (c=9) (n=7179)
10021011220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101 (c=10) (n=64284)
1001010211220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102 (c=11) (n=553929)
1001011021220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110 (c=12) (n=554253)
10010111021220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111 (c=13) (n=1663062)

Look at 1,001,010,211,220 (n=553929) and 1,001,011,021,220 (n=554253). They have the same number of digits, but the babushka 1,001,011,021,220 manages to pack in one more baby:


1001010211220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102 (c=11) (n=553929)
1001011021220 contains 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110 (c=12) (n=554253)

That happens in binary too:


10010111000 contains 1, 10, 11, 100, 101, 110, 111, 1000, 1001 (c=9) (n=1208)
10011101000 contains 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010 (c=10) (n=1256)

What happens in higher bases? Watch this space.

T4K1NG S3LF13S

It’s like watching a seed grow. You take a number and count how many 0s it contains, then how many 1s, how many 2s, 3s, 4s and so on. Then you create a new number by writing the count of each digit followed by the digit itself. Then you repeat the process with the new number.

Here’s how it works if you start with the number 1:

1

The count of digits is one 1, so the new number is this:

→ 11

The count of digits for 11 is two 1s, so the next number is:

→ 21

The count of digits for 21 is one 1, one 2, so the next number is:

→ 1112

The count of digits for 1112 is three 1s, one 2, so the next number is:

→ 3112

The count of digits for 3112 is two 1s, one 2, one 3, so the next number is:

→ 211213

What happens after that? Here are the numbers as a sequence:

1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 114213 → 31121314 → 41122314 → 31221324 → 21322314

That’s all you need, because something interesting happens with 21322314. The digit count is two 1s, three 2s, two 3s, one 4, so the next number is:

→ 21322314

In other words, 21322314 is what might be called a self-descriptive number: it describes the count of its own digits. That’s why I think this procedure is like watching a seed grow. You start with the tiny seed of 1 and end in the giant oak of 21322314, whose factorization is 2 * 3^2 * 13 * 91121. But there are many more self-descriptive numbers in base ten and some of them are much bigger than 21322314. A047841 at the Online Encyclopedia of Integer Sequences lists all 109 of them (and calls them “autobiographical numbers”). Here are a few, starting with the simplest possible:

22 → two 2s → 22
10213223 → one 0, two 1s, three 2s, two 3s → 10213223
10311233 → one 0, three 1s, one 2, three 3s → 10311233
21322314 → two 1s, three 2s, two 3s, one 4 → 21322314
21322315 → two 1s, three 2s, two 3s, one 5 → 21322315
21322316 → two 1s, three 2s, two 3s, one 6 → 21322316*
1031223314 → one 0, three 1s, two 2s, three 3s, one 4 → 10
31223314
3122331415 → three 1s, two 2s, three 3s, one 4, one 5
→ 3122331415
3122331416 → three 1s, two 2s, three 3s, one 4, one 6
→ 3122331416*

*And for 21322317, 21322318, 21322319; 3122331417, 3122331418, 3122331419.


And here’s what happens when you seed a sequence with a number containing all possible digits in base ten:

1234567890 → 10111213141516171819 → 101111213141516171819 → 101211213141516171819 → 101112213141516171819

That final number is self-descriptive:

101112213141516171819 → one 0, eleven 1s, two 2s, one 3, one 4, one 5, one 6, one 7, one 8, one 9 → 101112213141516171819

So some numbers are self-descriptive and some start a sequence that ends in a self-descriptive number. But that doesn’t exhaust the possibilities. Some numbers are part of a loop:

103142132415 → 104122232415 → 103142132415
104122232415 → 103142132415 → 104122232415
1051421314152619 → 1061221324251619 → 1051421314152619…
5142131415261819 → 6122132425161819 → 5142131415261819
106142131416271819 → 107122132426171819 → 106142131416271819


10512223142518 → 10414213142518 → 10512213341518 → 10512223142518
51222314251718 → 41421314251718 → 51221334151718 →
51222314251718

But all that is base ten. What about other bases? In fact, nearly all self-descriptive numbers in base ten are also self-descriptive in other bases. An infinite number of other bases, in fact. 22 is a self-descriptive number for all b > 2. The sequence seeded with 1 is identical in all b > 4:

1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 114213 → 31121314 → 41122314 → 31221324 → 2132231421322314

In bases 2, 3 and 4, the sequence seeded with 1 looks like this:

1 → 11 → 101 → 10101 → 100111 → 1001001 → 1000111 → 11010011101001… (b=2) (1101001[2] = 105 in base 10)
1 → 11 → 21 → 1112 → 10112 → 1010112 → 2011112 → 10111221011122… (b=3) (1011122[3] = 854 in base 10)
1 → 11 → 21 → 1112 → 3112 → 211213 → 312213 → 212223 → 1110213 → 101011213 → 201111213 → 101112213101112213… (b=4) (101112213[4] = 71079 in base 10)

In base 2 there are only two self-descriptive numbers (and no loops):

111 → three 1s → 111… (b=2) (111 = 7 in base 10)
1101001 → three 0s, four 1s → 1101001… (b=2) (1101001 = 105 in base 10)

So if you apply the “count digits” procedure in base 2, all numbers, except 111, begin a sequence that ends in 1101001. Base 3 has a few more self-descriptive numbers and also some loops:

2222… (b >= 3)
10111 → one 0, four 1s → 10111… (b=3)
11112 → four 1s, one 2 → 11112
100101 → three 0s, three 1s → 100101… (b=3)
1011122 → one 0, four 1s, two 2s → 1011122… (b=3)
2021102 → two 0s, two 1s, three 2s → 2021102… (b=3)
10010122 → three 0s, three 1s, two 2s → 10010122


2012112 → 10101102 → 10011112 → 2012112
10011112 → 2012112 → 10101102 → 10011112
10101102 → 10011112 → 2012112 → 10101102

A question I haven’t been able to answer: Is there a base in which loops can be longer than these?

103142132415 → 104122232415 → 103142132415
10512223142518 → 10414213142518 → 10512213341518 → 10512223142518

A question I have been able to answer: What is the sequence when it’s seeded with the title of this blog-post? T4K1NGS3LF13S is a number in all bases >= 30 and its base-30 form equals 15,494,492,743,722,316,018 in base 10 (with the factorization 2 * 72704927 * 106557377767). If T4K1NGS3LF13S seeds a sequence in any b >= 30, the result looks like this:

T4K1NGS3LF13S → 2123141F1G1K1L1N2S1T → 813213141F1G1K1L1N1S1T → A1122314181F1G1K1L1N1S1T → B1221314181A1F1G1K1L1N1S1T → C1221314181A1B1F1G1K1L1N1S1T → D1221314181A1B1C1F1G1K1L1N1S1T → E1221314181A1B1C1D1F1G1K1L1N1S1T → F1221314181A1B1C1D1E1F1G1K1L1N1S1T → G1221314181A1B1C1D1E2F1G1K1L1N1S1T → F1321314181A1B1C1D1E1F2G1K1L1N1S1T → F1222314181A1B1C1D1E2F1G1K1L1N1S1T → E1421314181A1B1C1D1E2F1G1K1L1N1S1T → F1221324181A1B1C1D2E1F1G1K1L1N1S1T → E1421314181A1B1C1D1E2F1G1K1L1N1S1T