## There are several ways to write the number $\dfrac{1}{2}$ as a sum of square reciprocals using distinct integers.

For instance, the numbers $\{2,3,4,5,7,12,15,20,28,35\}$ can be used:

$$\begin{align}\dfrac{1}{2} &= \dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2} + \dfrac{1}{5^2} +\\

&\quad \dfrac{1}{7^2} + \dfrac{1}{12^2} + \dfrac{1}{15^2} + \dfrac{1}{20^2} +\\

&\quad \dfrac{1}{28^2} + \dfrac{1}{35^2}\end{align}$$

In fact, only using integers between $2$ and $45$ inclusive, there are exactly three ways to do it, the remaining two being: $\{2,3,4,6,7,9,10,20,28,35,36,45\}$ and $\{2,3,4,6,7,9,12,15,28,30,35,36,45\}$.

How many ways are there to write $\dfrac{1}{2}$ as a sum of reciprocals of squares using distinct integers between $2$ and $80$ inclusive?

### To find the number of ways to write $\dfrac{1}{2}$ as a sum of reciprocals of squares using distinct integers between $2$ and $80$ inclusive, we can use a combination of brute force and a recursive approach.

First, let’s define a function called `find_ways` that takes in a target fraction, lower and upper bounds for the integers, and a current sum of squares.

“`python

def find_ways(target, lower, upper, current_sum):

# Base case: if the target is reached, return 1

if target == 0:

return 1

# Base case: if the target goes below 0 or the current integer exceeds the upper bound, return 0

if target < 0 or lower > upper:

return 0

# Recursive case: try adding the next integer square to the sum

include_next = find_ways(target – 1 / (lower ** 2), lower + 1, upper, current_sum + [lower])

# Recursive case: try excluding the next integer square from the sum

exclude_next = find_ways(target, lower + 1, upper, current_sum)

# Return the total number of ways by summing the include and exclude cases

return include_next + exclude_next

“`

Using this function, we can call `find_ways(1/2, 2, 80, [])` to find the number of ways to write $\dfrac{1}{2}$ as a sum of reciprocals of squares using distinct integers between $2$ and $80$ inclusive.

“`python

num_ways = find_ways(1/2, 2, 80, [])

print(num_ways)

“`

The output will be the total number of ways.

(Note: This solution may require a significant amount of time and computational resources for larger inputs, as it uses a brute force approach. Implementations of dynamic programming or optimizations such as memoization could significantly improve the efficiency of this solution.)

##### More Answers:

Maximum-sum SubsequenceSub-triangle Sums

A Preference for A5