Can you solve the Leonardo da Vinci riddle? - Tanya Khovanova
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Let’s Begin…
You’ve found Leonardo da Vinci’s secret vault, secured by a series of combination locks. Fortunately, your treasure map has three codes: 1210, 3211000, and… hmm. The last one appears to be missing. Can you figure out the last number and open the vault? Tanya Khovanova shows how.
Additional Resources for you to Explore
To learn more about 10-digit autobiographical numbers, check out Martin Gardner's Mathematical Circus.
Interestingly, we can expand on autobiographical numbers. Consider this: if there is a notion of an autobiography of a number, then it would be logical to expect there to be a notion of a biography of a number. What would be the logical candidate for a biography of a number? Let us say that given a number N, its biography is another number M such that the first digit of M is the number of zeroes in N, the second digit of M is the number of ones in N and so on.
For example, it is possible to have two numbers that are biographies of each other: 130 and 1101. Can you find two 10-digit numbers that are biographies of each other?
If you want to find out more about the biographies of numbers, check out the educator’s "A Story of Storytelling Numbers."
You can also explore the educator’s math blog here.
There are many interesting sequences that describe themselves. For example, the Golomb sequence, a(n), is lexicographically the earliest non-decreasing sequence of positive integers such that a(n) is the number of times that n occurs in the sequence. It has to start with 1: a(1) = 1. That means we have 1 one. Therefore, a(2) can't be 1. We pick the next smallest number available, so a(2) =2. Now we have to have 2 twos, so we need one more two. Thus a(3) = 2. The last equation says that the sequence has to have 2 threes. As we already used all ones and twos, we have a(4) = a(5) = 3. The first few terms of the sequence are:
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12.
This sequence is sequence A001462 in the Online Encyclopedia of Integer Sequences.
Interestingly, we can expand on autobiographical numbers. Consider this: if there is a notion of an autobiography of a number, then it would be logical to expect there to be a notion of a biography of a number. What would be the logical candidate for a biography of a number? Let us say that given a number N, its biography is another number M such that the first digit of M is the number of zeroes in N, the second digit of M is the number of ones in N and so on.
For example, it is possible to have two numbers that are biographies of each other: 130 and 1101. Can you find two 10-digit numbers that are biographies of each other?
If you want to find out more about the biographies of numbers, check out the educator’s "A Story of Storytelling Numbers."
You can also explore the educator’s math blog here.
There are many interesting sequences that describe themselves. For example, the Golomb sequence, a(n), is lexicographically the earliest non-decreasing sequence of positive integers such that a(n) is the number of times that n occurs in the sequence. It has to start with 1: a(1) = 1. That means we have 1 one. Therefore, a(2) can't be 1. We pick the next smallest number available, so a(2) =2. Now we have to have 2 twos, so we need one more two. Thus a(3) = 2. The last equation says that the sequence has to have 2 threes. As we already used all ones and twos, we have a(4) = a(5) = 3. The first few terms of the sequence are:
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12.
This sequence is sequence A001462 in the Online Encyclopedia of Integer Sequences.
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