Practice Questions

Define a palindromic number.

Apply the Collatz conjecture rule to the starting number 20. Calculate the next three numbers in the sequence.

Examine the list of numbers: [412, 214, 421, 124, 241]. Apply the definition of a supercell to identify all supercells in this list.

State the Kaprekar constant for 4-digit numbers.

List three different 4-digit palindromic numbers.

List all the 2-digit numbers whose digits add up to 5.

Evaluate what happens when the Kaprekar process is applied to a 4-digit number where all digits are the same, for example, 3333.

Justify why it is impossible to create a 3-digit number whose digits are all odd, and whose digit sum is an even number.

Calculate the next palindromic number that comes immediately after 8998.

Identify the smallest 3-digit number.

Explain what a 'supercell' is in a row of numbers. Identify the supercell in the list: [45, 80, 62].

What is the digit sum of the number 504?

Five children with heights 152 cm, 145 cm, 155 cm, 148 cm, and 150 cm stand in a line in this specific order. Apply the rule where each child says the number of their taller immediate neighbors. Determine the sequence of numbers the children will say.

Name the two main rules applied in the Collatz conjecture sequence generation.

Describe the first step of the 'reverse-and-add' method to generate a palindrome. Use the number 62 as an example.

Describe the complete procedure to find the Kaprekar constant (6174) starting with a 4-digit number. Use the number 3524 to illustrate the first cycle of the procedure.

Identify all the palindromic numbers from the following list and explain your reasoning for each choice: 232, 445, 88, 707, 1231, 6006.

Apply one step of the Kaprekar's process to the number 5295. Calculate the result of the first subtraction.

Calculate the number of steps it takes for the 3-digit number 751 to reach the Kaprekar constant for 3-digit numbers (495).

Calculate the smallest 3-digit number whose digits sum to 17.

Analyze the given grid. Fill the three blank cells with unique 2-digit numbers, such that there are exactly three supercells in the final 6-cell grid. [50, ___, 80, ___, 90, ___]

Apply the "reverse-and-add" method to the number 78. Calculate the steps required until a palindrome is formed.

Analyze the number line below where the marks are equally spaced. Calculate the values of A and B.

|----|----A----|----|----|----B----| 3500 5000

Evaluate the claim: "The reverse-and-add process for any 2-digit number ending in 9 will always result in a palindrome in 2 steps or less." Provide an example that supports or refutes this claim.

Justify why in a 1x9 grid filled with distinct numbers, it is impossible to have 9 supercells.

Design a 1x7 grid using distinct 3-digit numbers such that it contains exactly 3 supercells, and the middle cell is not a supercell. Justify why your arrangement meets the conditions.

A student claims, "For the Collatz Conjecture, if a starting number is a power of 2, like 32, the sequence will only contain even numbers until it reaches 1." Critique this statement. Is it always true? Provide your reasoning.

Create a 5-digit palindromic number that satisfies the following conditions:

1. It is an even number.
2. The sum of its digits is 20.
3. The digit in the hundreds place is the largest digit in the number. Present the number and show that it meets all conditions.

Identify how many 1-digit numbers and 2-digit numbers exist in our number system.

Explain the concept of 'digit sum'. Find the digit sum for the following three numbers: 241, 58, and 1099. State which number has the smallest digit sum.

Formulate a rule about the digit sum of any number that is a multiple of 9. Test your rule with three different multiples of 9 (e.g., 27, 198, 4536). Justify why this pattern might occur.

Explain why the smallest number in a row of unique numbers can never be a supercell.

a) Calculate the complete sequence of steps for the number 3051 to reach the Kaprekar constant, 6174. b) Analyze and explain why the Kaprekar process requires the number to have at least two different digits.

List the first four terms of the Collatz sequence that starts with the number 6.

For 5 children of different heights, evaluate if the sequence 2, 1, 0, 1, 2 is possible. Justify your answer with a diagram or a clear explanation of the height arrangement required.

Analyze the following clues to solve the puzzle and find the 5-digit number.

1. I am a 5-digit palindromic number.
2. I am an even number.
3. The sum of my digits is 21.
4. My hundreds digit is three times my thousands digit.

Analyze the 2D grid of numbers below. A cell is a supercell if its value is greater than all its immediate neighbors (left, right, top, bottom).

+-----+-----+-----+ | 450 | 500 | 350 | +-----+-----+-----+ | 400 | 200 | 300 | +-----+-----+-----+ | 380 | 250 | 280 | +-----+-----+-----+

a) Identify all the supercells in the current grid. b) Demonstrate how you can change the value of exactly one cell to create a total of four supercells.

Analyze the conditions to find the largest 4-digit odd number whose digits sum to 25.

Create an example where the difference between two 5-digit numbers is a 2-digit number. Justify why the difference between a 5-digit number and a 4-digit number can never be a 5-digit number.

The Kaprekar process for 4-digit numbers uses subtraction. Propose a similar process using addition (i.e., add the largest and smallest numbers formed by the digits). Evaluate if this new process leads to a repeating number or a constant for the starting number 2024.

Calculate the full Collatz sequence for the starting number 19. Determine the length of the sequence (number of terms until it reaches 1).