Hello, young mathematicians! Today, we have a fun puzzle to solve. Imagine you have seven different positive integers. They are like seven different friends, each with their own unique personality. We want to line them up in a row, but there’s a catch. The sum of any three neighboring friends must be more than 15. What’s the smallest possible sum of all seven friends? Let’s find out!

Let’s give our seven integer friends names: $a$, $b$, $c$, $d$, $e$, $f$, and $g$. The rule is that any three neighboring friends, say $a$, $b$, and $c$, must have a sum greater than or equal to 16. We can write this as $(a+b+c) \geq 16$.

Now, let’s say the total sum of all seven friends is $s$. We can write this as $s = (a+b+c) + (d+e+f) + g$. According to our rule, this should be greater than or equal to $32 + g$.

But wait! We can also group our friends differently. What if we group them as $s = (a+b+c) + d + (e+f+g)$? According to our rule, this should also be greater than or equal to $32 + d$.

And there’s yet another way to group them: $s = a + (b+c+d) + (e+f+g)$. This should be greater than or equal to $32 + a$.

Now, remember that $a$, $d$, and $g$ are all different positive integers. They’re like the smallest, second smallest, and third smallest kids in a class. So, the smallest they can be are 1, 2, and 3.

So, we have $s \geq 32 + 1$, $s \geq 32 + 2$, and $s \geq 32 + 3$. When $s \geq 35$, all conditions are satisfied.

Therefore, the smallest possible sum of our seven integer friends is 35. And there you have it! Isn’t it fun to play with numbers? Stay tuned for more exciting math adventures!