Explain why all 7-digit pointy numbers that are divisible by 3 must have their first digit divisible by 3.

A 7-digit pointy number is a number with digits arranged such that they form a 'point' shape when connected. For example, 123321 or 135753 are 7-digit pointy numbers. Let's denote a 7-digit pointy number as ABCDCBA, where each letter represents a digit. Now, let's break down why such numbers must have their first digit divisible by 3: Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. Symmetry: In a 7-digit pointy number, if you split it at the middle digit, you'll notice that the left side is a mirror image of the right side. For example, in 123321, 123 is the mirror image of 321. Sum of Digits: Since the number is symmetrical, the sum of digits on the left side must be equal to the sum of digits on the right side. First Digit: The first digit, A, is on the left side. If A is not divisible by 3, changing it will affect the sum of the left side digits but not the right side. Preserving Divisibility: If we change A, we would need to change the corresponding digit on the right side to maintain symmetry. However, if A wasn't divisible by 3 to begin with, changing it to a digit divisible by 3 would alter the divisibility by 3 condition. Therefore, to maintain the divisibility by 3 condition and symmetry in the number, the first digit (A) must be divisible by 3.