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Ok so I recently came across the spell plant growth, which says for every 1ft moved you require 4ft instead.

This prompted me to look in the rules, and I found out that you can use any amount of your movement you want each round, this really confused me since I always thought in squares or hexagons.

Ultimately my question is not about this spell at all but about how does this strange movement work? If you move 17ft do you only move 3 squares because you could not complete the last few feet or do you move 4 squares rounding up your movement?

If you only move 4ft each round do you just not move at all or do you just jump up one square?

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In 5e grids are an optional rule, the rules don't assume you are playing on a grid leading to awkward situations. The movement rules are no help, they just say "you can move a distance up to your speed". The difficult terrain rules say "Every foot of movement in difficult terrain costs 1 extra foot. This rule is true even if multiple things in a space count as difficult terrain." but again don't mention what to do with grids.

In the DMG the section called "Tactical Maps" covers how to play with grids:

You can draw tactical maps with colored markers on a wet-erase vinyl mat with 1-inch squares, on a large sheet of paper, or on a similar flat surface. Preprinted poster-sized maps, maps assembled from cardboard tiles, and terrain made of sculpted plaster or resin are also fun.

The most common unit for tactical maps is the 5-foot square, and maps with grids are readily available and easy to create. However, you don’t have to use a grid at all. You can track distances with a tape measure, string, craft sticks, or pipe cleaners cut to specific lengths. Another option is a play surface covered by 1-inch hexagons (often called hexes), which combines the easy counting of a grid with the more flexible movement of using no grid. Dungeon corridors with straight walls and right angles don’t map easily onto hexes, though.

In the rules for Movement there's a section on the Playing on a Grid variant rule which offers a little more information:

Rather than moving foot by foot, move square by square on the grid. This means you use your speed in 5-foot segments. This is particularly easy if you translate your speed into squares by dividing the speed by 5. For example, a speed of 30 feet translates into a speed of 6 squares.

...

To enter a square, you must have at least 1 square of movement left

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If a square costs extra movement, as a square of difficult terrain does, you must have enough movement left to pay for entering it. For example, you must have at least 2 squares of movement left to enter a square of difficult terrain.

Divide your movement into squares, then pay 3 extra squares of movement every time you move 1 square. If you don't have 4 squares total available to pay, you can't move.

If your character has 17 feet of speed, then divide that into 3.4, then round it down to 3 squares of movement. You need 4 squares to move, so in this example you can't.

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In this situation, you cannot move

When moving on a grid, movement is measured in squares; to enter a space that costs four times as much movement, you must have four squares of movement available.

The rules on Playing on a Grid state:

Squares. Each square on the grid represents 5 feet.

Speed. Rather than moving foot by foot, move square by square on the grid. This means you use your speed in 5-foot segments. This is particularly easy if you translate your speed into squares by dividing the speed by 5. For example, a speed of 30 feet translates into a speed of 6 squares.

[...]

Entering a Square. To enter a square, you must have at least 1 square of movement left, even if the square is diagonally adjacent to the square you’re in. (The rule for diagonal movement sacrifices realism for the sake of smooth play. The Dungeon Master’s Guide provides guidance on using a more realistic approach.)

If a square costs extra movement, as a square of difficult terrain does, you must have enough movement left to pay for entering it. For example, you must have at least 2 squares of movement left to enter a square of difficult terrain. [...]

So from this we can tackle your example scenario. There is a terrain that requires four feet of movement for each foot travelled. This is similar to difficult terrain which would require two squares of movement but now we require four squares of movement. Now we just have to calculate our speed in terms of squares.

Your speed is three squares, so you cannot move

When calculating our speed in terms of squares, we divide our speed by five, and because of the Round Down rule, we round down.

[...] Whenever you divide a number in the game, round down if you end up with a fraction, even if the fraction is one-half or greater.

So in this particular scenario we have 17 feet of movement which would be 17/5 = 3 squares of movement. Because the terrain created by plant growth would require four squares of movement, we cannot enter or move through it.

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Play Without Map and Minis - Theater of the Mind

Some tables play without a grid. In 5e, unlike some other editions, the grid is an optional rule, not a core rule. It is so common that we often take for it for granted that people are using grid squares. 5e can be played entirely "theatre of the mind" without maps and minis at all. I personally have only played theatre of the mind with other systems; but there is nothing special about minis and map.

Play Without A Grid - Gridless

Some tables play instead a rulers (sometimes templates) and math, and you really can move any distance less than your total movement. I've see some terrain crafters on YouTube that talk about the freedom they feel going "gridless". This would have you move 4/5"ish on the map, which isn't a big deal at all.

With a Grid

There are four ways I've seen at people use the grid for odd increments of movement:

Round Down

If you can't move enough to leave the square, you effectively didn't move. Any movement under 5ft just "doesn't count" and is just shifting around in your square. I believe this is what a strict reading of the optional rules would have you do as Exempt-Medic's answer explains. This would have you not move at all.

Center "of Mass"

If you control a space 5ft, the midpoint is where you "are" so if you move more than 2.5ft, you're effectively in the next square and it is effectively rounding to the movement to the nearest 5ft. This seems the most "fair" but I don't think it is as common as the previous one. This is also similar to what the optional rules say about when damage from an area effect should apply, that if the majority of the square is covered by the effect a creature in the square is affected, but if less than half the square is covered, a creature in the square is not affected. This would have you move up one square.

Round Up

I don't think I've ever seen this played for movement, but I have seen people play it for area effects because they like "bigger fireballs" and what not. I don't think this is a likely answer for movement less than 5ft. This would also have you move up one square.

Divide the Grid

You can live with a character not quite in the grid system. This would have you move over 4/5th a square, or I've seen some just handwave "It'll takes two turns movement to leave get there, using your full movement each turn" because they don't the mini not clearly in a grid box.

Whatever You Do, Keep it Consistent

How a DM chooses to handle it, they should be consistent about it when it comes up. What you do for a PC, you do for monsters and vice versa. They can change how they handle the uneven movement amounts if they choose, but it should be communicated to table and from then on they should do it the new way for any of the rate times it comes up. That said, it is a rare thing and many DMs won't run into it, because most of the rules are designed around numbers divisible by 5 specifically to be grid-friendly.

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