# Difference between revisions of "Euler problems/81 to 90"

## Problem 81

Find the minimal path sum from the top left to the bottom right by moving right and down.

Solution:

```problem_81 = undefined
```

## Problem 82

Find the minimal path sum from the left column to the right column.

Solution:

```problem_82 = undefined
```

## Problem 83

Find the minimal path sum from the top left to the bottom right by moving left, right, up, and down.

Solution:

```problem_83 = undefined
```

## Problem 84

In the game, Monopoly, find the three most popular squares when using two 4-sided dice.

Solution:

```problem_84 = undefined
```

## Problem 85

Investigating the number of rectangles in a rectangular grid.

Solution:

```problem_85 = undefined
```

## Problem 86

Exploring the shortest path from one corner of a cuboid to another.

Solution:

```problem_86 = undefined
```

## Problem 87

Investigating numbers that can be expressed as the sum of a prime square, cube, and fourth power?

Solution:

```import List

problem_87 = length expressible
where limit = 50000000
squares = takeWhile (<limit) (map (^2) primes)
cubes   = takeWhile (<limit) (map (^3) primes)
fourths = takeWhile (<limit) (map (^4) primes)
choices = [[s,c,f] | s <- squares, c <- cubes, f <- fourths]
unique  = map head . group . sort
expressible = filter (<limit) . unique . map sum \$ choices
```

## Problem 88

Exploring minimal product-sum numbers for sets of different sizes.

Solution:

```problem_88 = undefined
```

## Problem 89

Develop a method to express Roman numerals in minimal form.

Solution:

```problem_89 = undefined
```

## Problem 90

An unexpected way of using two cubes to make a square.

Solution:

```problem_90 = undefined
```