Imagine you’ve stumbled upon a secret vault belonging to the legendary Leonardo da Vinci. It’s locked with a series of combination locks, but luckily, you have a treasure map with some clues. The map gives you two codes: 1210 and 3211000. However, the last code is missing, and it’s up to you to figure it out!
The first two numbers are called autobiographical numbers. These numbers are special because they describe themselves. Each digit in the number tells you how many times the digit that matches its position appears in the number. For example, in the number 1210, the first digit ‘1’ means there is one ‘0’, the second digit ‘2’ means there are two ‘1s’, and so on.
The last lock needs a 10-digit autobiographical number. There’s only one such number, and you need to find it. Let’s break it down step by step.
First, notice that if you add up all the digits in an autobiographical number, it equals the total number of digits in the number. For 1210, the sum is 4, which matches the number of digits. So, for a 10-digit autobiographical number, the digits must add up to 10.
This means the number can’t have too many large digits. For instance, if it had a ‘6’ and a ‘7’, that would mean some digits appear 6 and 7 times, which is more than 10 digits in total. Therefore, only one digit can be greater than 5, if any.
There will be zeroes in the positions for digits that aren’t used. This means our number must have at least three zeroes, so the first digit must be 3 or more.
The first digit counts the number of zeroes, and each digit after it counts how many times a specific non-zero digit appears. If we add up all the digits except the first one, we get the total number of non-zero digits in the sequence, including the leading digit.
For example, in the code 1210, adding 2 and 1 gives us 3. Subtracting 1, we find there are two non-zero digits after the first digit. This tells us that the total number of non-zero digits after the first digit equals the sum of these digits minus one.
To make the sum exactly 1 more than the number of non-zero digits, one of the digits must be a 2, and the rest must be 1s. There can only be two 1s; otherwise, we’d need larger numbers like 3 or 4 to count them.
Now, we know the leading digit is 3 or more to count the zeroes, a 2 to count the 1s, and two 1s—one to count the 2s and another to count the leading digit.
To find the leading digit, we subtract the sum of 2 and the two 1s (which is 4) from 10, leaving us with 6. So, we arrange them: 6 zeroes, 2 ones, 1 two, 0 threes, 0 fours, 0 fives, 1 six, 0 sevens, 0 eights, and 0 nines.
With this arrangement, the safe opens, revealing Da Vinci’s long-lost autobiography!
Research and find other examples of autobiographical numbers besides 1210 and 3211000. Create a poster that explains how each number describes itself. Present your findings to the class and explain why these numbers are unique.
Using the concept of autobiographical numbers, design a riddle similar to the Leonardo da Vinci riddle. Include a series of clues that lead to a solution. Share your riddle with classmates and see if they can solve it.
Work in groups to create a puzzle that involves finding a 10-digit autobiographical number. Use the rules and patterns discussed in the article. Exchange puzzles with another group and try to solve each other’s challenges.
Leonardo da Vinci was both an artist and a mathematician. Research how math is used in art, particularly in da Vinci’s work. Create a piece of art that incorporates mathematical concepts, and explain the math behind your creation.
Write a short story or comic strip that involves a character solving a mystery using autobiographical numbers. Use the story to illustrate how these numbers can be applied in a real-world scenario. Share your story with the class.
You’ve discovered a secret vault belonging to Leonardo Da Vinci, secured by a series of combination locks. Fortunately, your treasure map provides three codes: 1210, 3211000, and… it seems the last one is missing. You’ll need to figure it out on your own.
The first two numbers are known as autobiographical numbers. This special type of number describes itself: each digit indicates how many times the digit corresponding to that position occurs within the number. For instance, the first digit indicates the quantity of zeroes, the second digit indicates the number of ones, the third digit indicates the number of twos, and so on.
The last lock requires a 10-digit number, and there is exactly one ten-digit autobiographical number. What is it? Take a moment to think about it if you’d like!
Randomly trying different combinations would take a long time, so let’s analyze the autobiographical numbers we have to find patterns. If we add all the digits in 1210 together, we get 4, which is the total number of digits. This makes sense since each digit tells us how many times a specific digit occurs within the total. Therefore, the digits in our ten-digit autobiographical number must add up to ten.
This also means the number can’t have too many large digits. For example, if it included a 6 and a 7, then some digit would have to appear 6 times, and another digit 7 times, resulting in more than 10 digits. We can conclude that there can be no more than one digit greater than 5 in the entire sequence. Thus, out of the digits 6, 7, 8, and 9, only one—if any—will be included.
Additionally, there will be zeroes in the positions corresponding to the numbers that aren’t used. We now know that our number must contain at least three zeroes, which also means that the leading digit must be 3 or greater.
While the first digit counts the number of zeroes, every digit after it counts how many times a particular non-zero digit occurs. If we add together all the digits besides the first one—keeping in mind that zeroes don’t increase the sum—we get a count of how many non-zero digits appear in the sequence, including that leading digit.
For example, using the first code, we get 2 plus 1 equals 3 digits. If we subtract one, we find that there are two non-zero digits after the first digit. This leads us to an important conclusion: the total quantity of non-zero digits that occur after the first digit is equal to the sum of these digits minus one.
To achieve a distribution where the sum is exactly 1 greater than the number of non-zero positive integers being added together, one of the addends must be a 2, and the rest must be 1s. How many 1s? It turns out there can only be two—any more would require additional digits like 3 or 4 to count them.
Now we have the leading digit of 3 or greater counting the zeroes, a 2 counting the 1s, and two 1s—one to count the 2s and another to count the leading digit.
Next, we need to determine the leading digit. Since we know that the 2 and the two 1s sum to 4, we can subtract that from 10 to get 6. Now, we can arrange them: 6 zeroes, 2 ones, 1 two, 0 threes, 0 fours, 0 fives, 1 six, 0 sevens, 0 eights, and 0 nines.
The safe swings open, revealing Da Vinci’s long-lost autobiography.
Number – A mathematical object used to count, measure, and label. – In algebra, we often solve equations to find the unknown number.
Digits – The symbols used to represent numbers, typically 0 through 9. – The number 345 has three digits: 3, 4, and 5.
Count – To determine the total number of items in a set. – We count the number of solutions to the equation to see how many values satisfy it.
Zeroes – The digits that represent the number zero, often used to indicate the absence of a value. – In the equation x^2 – 4 = 0, the zeroes are the values of x that make the equation true.
Sum – The result of adding two or more numbers together. – The sum of 7 and 5 is 12.
Leading – Referring to the first or most significant digit in a number. – In the number 456, the leading digit is 4.
Puzzle – A problem designed to test ingenuity or knowledge, often involving numbers or patterns. – Solving algebraic puzzles can help improve problem-solving skills.
Pattern – A repeated or regular arrangement of numbers, shapes, or other elements. – Recognizing the pattern in a sequence can help predict the next number.
Code – A system of symbols used to represent information, often used in mathematics to simplify expressions. – In algebra, we use a code of variables to represent unknown quantities.
