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Is It a Digit?

In this engaging puzzle, you must place the digits 0 through 4 into five boxes labeled 0 to 4, adhering to specific rules. Each box's digit must equal the number of times that digit appears in the boxes. Your task is to find all valid combinations while proving their uniqueness. Explore an example of incorrect placement to understand the constraints better. Once solved, document your problem-solving approach, list all solutions, and provide a detailed proof of their uniqueness. Join in on this mathematical challenge and test your logic skills!

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Is It a Digit?

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  1. Is It a Digit? IMP Solve It!

  2. There are five empty boxes shown here labeled 0 through 4. 0 1 2 3 4

  3. Your task is to put a digit 0 through 4 inside each of the boxes so that certain conditions hold: • The digit you put in the box labeled “0” must be the same as the number of 0’s you use. • The digit you put in the box labeled “1” must be the same as the number of 1’s you use. • The digit you put in the box labeled “2” must be the same as the number of 2’s you use, and so on… 3 4 0 1 2

  4. Additional Information What to do What Not to Do 2 2 3 1 2 2 You are allowed to use the same digit more than once. You may want to make several copies of the set of boxes in order to try the various combinations of digits. • Here is an example of an incorrect way to fill the boxes. • This is incorrect for many reasons. For instance, there is a 1 in the box labeled “2”, but there is more than one 2 in the boxes…..

  5. Once you have solved the problem in Is it a Digit?, your task is to prove that you have all the solutions. Write Up • Problem Statement: Explain the problem from Is it a Digit? • Process: Based on your notes, describe how you went about finding all the solutions to Is it a Digit? And how you decided that you had them all. • Solutions: List all solutions you found for the Is it a Digit? Then write a careful and detailed proof that there are no solutions to Is it a Digit? other than those listed. • Evaluation • Self-assessment POW 1 A Digital Proof

  6. Warning!!!!!! • The answer(s) follow on the following slide.

  7. The order of the numbers should be • 2 1 2 0 0 • The proof that the solution is unique consists fo eliminating all other possibilities. However, there are so many possibilities (55 or 3125) the other cases need to be eliminated in an organized and systematic way. • However, once 3 and 4 are eliminated, a proof could be made by showing the 33 or 27 possibilities. • Also a proof can be shown that the digits must be equal to 5.

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