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Yeah, that makes sense. The tunnel is awful small to try a fireball too, then. So that's out.
Alie tries to cast her Confusion spell, blasting everyone in the front cart.
DC 24 Will to resist.

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During this round, you cannot see the cart, just the tracks where they will go.
-Posted with Wayfinder

GM Bold Strider |

Seems like no one is going to sabotage the tracks.
Random Event: 1d6 ⇒ 2
You hear a giant thwock! from the up around the bend. As you come barreling down the tracks, you see that the guards lodged an axe into one of the supports, cutting the ropes. A giant beam of wood lays across the tracks. Your cart rams into the lumber, slowing your speed by 10'/rd. and forcing Jaggar to stabilize the cart.
Ride Check DC 15 or everyone takes 1d4 non-lethal damage.
@Alie: Extremely Violent Motion concentration check on the spell. DC 20+Spell Level
Initiative - Cart: Speed=60'/rd.; 190' travelled
1st: Jaggar - Driving Cart; Mathais;
2nd: Woman @ 7 (On Cart); Guards (#1-4 @ 4 - On Cart) [50'/rd.] [430']
3rd: Jarlgur; Alie; Maaarv

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"If we have to chase them down, we need to move quickly," Alie mutters as she casts Haste.
Remember all your Haste bonuses, please.

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Ride: 1d20 + 5 ⇒ (15) + 5 = 20
Suddenly, Maaarv spreads his wings and leaps up and out of the cart at a 45 degree angle (up 15, forward 15). He then flies along the track (60 feet).
So, I should be at 265' along the track. Following the track, I will double-move each round (120 feet) until I'm able to catch up with the other cart.

GM Bold Strider |

Since you are traveling through mineshafts and whipping around corners. I'm going to need a Fly DC 25 check every round to stay aloft as you are pivoting and doubling back around switchbacks at around 15 mph in a cramped mineshaft.
The Ride checks are only for Jaggar. Using your Ride check as your Fly check with the appropriate modifier.

GM Bold Strider |

Jaggar stabilizes the cart just before it slams into the lumber, causing the metal contraption to jostle significantly, but no one is hurt.
He maintains the speed of the vehicle, but can't seem to accelerate it as Alie speeds everyone up. Maaarv takes flight and wings off down the tracks.
Initiative - Cart: Speed=60'/rd.; 190' travelled
1st: Jaggar - Driving Cart; Mathais;
2nd: Woman @ 7 (On Cart); Guards (#1-4 @ 4 - On Cart) [50'/rd.] [430']
3rd: Jarlgur; Alie; Maaarv (Fly-265')

Signboy77 |

Oh, I was thrown off by the 'everyone' in the statement. I mistakenly thought it meant everyone needed to make the check. In that case, my Fly check is 31. Fly mod is +16 while under the effects of my mutagen.

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Mathais will try to aid Jagger's driving
Aid!: 1d20 ⇒ 6
But his hand slips while trying to help and does nothing :(

GM Bold Strider |

2d20 ⇒ (17, 3) = 20
Random Event: 1d6 ⇒ 3
Jaggar and Mathais continue to try and keep the mine cart from becoming smashed waste on the bottom of the various pits that the tracks traverse as they think they are making ground on the cart ahead.
Initiative - Cart: Speed=60'/rd.; 190' travelled
1st: Jaggar - Driving Cart; Mathais;
2nd: Woman @ 7 (On Cart); Guards (#1-4 @ 4 - On Cart) [50'/rd.] [480']
3rd: Jarlgur; Alie; Maaarv (Fly-265')

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Fly: 1d20 + 16 ⇒ (1) + 16 = 17
Maaarv is unable to navigate the labyrinthine nature of the tunnels and ends up falling to the ground.
Acrobatics to negate first 10 feet of damage: 1d20 + 5 ⇒ (2) + 5 = 7
Damage: 1d6 ⇒ 5
(Not sure how to do damage for a 15' fall. CRB seems to only indicate increments of 10 feet)
As he falls to the ground, Maaarv stands up immediately to prepare to fly again.

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Alie, too will try and aid the driver of their cart.
DM, I'm not sure what to roll to Aid. I'll just roll a d20 and you can apply any modifiers, if you don't mind.
Aid roll 1d20 ⇒ 6

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Jaggar continues to push the cart faster...
Ride1: 1d20 + 6 ⇒ (16) + 6 = 22
Ride2: 1d20 + 6 ⇒ (6) + 6 = 12
Ride Haster: 1d20 + 6 ⇒ (9) + 6 = 15
"Are we getting any closer for ranged targeting?" Jaggar asks his companions.

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The winged alchemist, in his tireless pursuit, doesn't allow something like slamming into a tunnel wall keep him down. He presses forward, though a bit more cautiously at the moment.
Keep double-moving but since I am moving at half speed, then I only move forward 60 feet along the track.

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I assume that we can't see the other vehicle ahead of us with the twists and turns. If so, I'll not take an action while we try to catch up.
In the event the group is able to see ahead of them at any point to see the cart before then, Alie will cast Silent Image of a small cave in front of the cart, hoping the driver will brake.

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I can't do anything at all, so I'll pass on actions, unless we get within 50' or if I'm needed to aid Jagger (straight d20 roll, tho :( )

GM Bold Strider |

@Alie: I like that. Very ingenius.
The woman in the lead cart can be heard yelling. "SPEED UP, YOU DOLT! THEY WILL CATCH US! WE MUST REPORT TO MY BROTHER!"
Ride: 1d20 + 5 ⇒ (6) + 5 = 11
Ride: 1d20 + 5 ⇒ (6) + 5 = 11
Alie spots a stretch of track through the caves and throws up a Silent Image of a cave wall across the track, causing the men to dive down and the driver to slam on the brakes, slowing their speed by 20'/rd. almost immediately.
1d3 ⇒ 2
Initiative - Cart: Speed=60'/rd.; 250' travelled
1st: Jaggar - Driving Cart; Mathais;
2nd: Woman @ 7 (On Cart); Guards (#1-4 @ 4 - On Cart) [30'/rd.] [530']
3rd: Jarlgur; Alie; Maaarv (Fly-325')

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Marvelous idea, Alie.
Fly: 1d20 + 16 ⇒ (3) + 16 = 19
Still finding it difficult to navigate through the tunnel, Maaarv again slams into the side of the tunnel, taking more damage.
Damage: 1d6 ⇒ 6
"Oh hey! They're coming up quick!" Maaarv observes of the group's minecart. He decides it may be easier to simply ride in the cart. He descends down and settles back in to the cart.
Not sure if I will need a fly check. The cart is only 5 feet behind me at this point. Normally, a fly check is not needed to descend, but maybe one is needed to land in a moving minecart.
Fly Check: 1d20 + 16 ⇒ (12) + 16 = 28

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"Keep pushing it! We are closing on them, I think. Oh, and that cave in is fake," Alie whispers to the others in the cart.
Alie will take a round off, pondering what other objects might make sense to appear in this tunnel in front of the cart.

GM Bold Strider |

The guardsmen try to get the mine cart back up to speed as the woman berates them for their incompetence.
"HOW CAN YOU LET A SIMPLE ILLUSION WORK ON YOU?!? ARE YOU AN IDIOT?!" can be heard echoing down the mine shaft.
Ride: 1d20 + 5 + 2 ⇒ (7) + 5 + 2 = 14
Ride: 1d20 + 5 + 2 ⇒ (12) + 5 + 2 = 19
Initiative - Cart: Speed=80'/rd.; 320' travelled
1st: Jaggar - Driving Cart; Mathais;
2nd: Woman @ 7 (On Cart); Guards (#1-4 @ 4 - On Cart) [50'/rd.] [580']
3rd: Jarlgur; Alie; Maaarv

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Alie holds her action until she saw an opening and fired away at any of the targets with a Magic Missile spell.
If she can see any of them, she will blast away. If so, damage 5d4 + 5 ⇒ (2, 4, 2, 1, 2) + 5 = 16.

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Mathais tries aiding Jagger's driving
Aid!: 1d20 ⇒ 12
And SUCCEEDS!

GM Bold Strider |

With Mathais' help, Jaggar is able to speed the cart up faster, however it begins to rumble and shake. Jaggar isn't sure he can keep this speed up for long without some injury to its occupants.
@Jaggar: You are at max speed. Now you only need to make one Ride check per round to maintain the speed. Better hope they are good.
Alie blasts one of the guards with her Magic Missiles. As it slams into his chest, he falls backwards over the side of the speeding mine cart.
"Ahhhhhhhhhhhhhhhhhhhhhhhh..." screams the man as he plummets down. A sickening crunch can be heard as his head and neck collide with one of the jutting platforms beneath them.
2d20 ⇒ (14, 20) = 34
"DO SOMETHING ABOUT THEM!" yells the woman as the guard driving gets the cart moving faster.
Initiative - Cart: Speed=100'/rd.; 420' travelled
1st: Jaggar - Driving Cart; Mathais;
2nd: Woman @ 7 (On Cart); Guards (#1-3 @ 4 - On Cart) [70'/rd.] [630']
3rd: Jarlgur; Alie; Maaarv

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Assuming the cart can close to within 190', Alie fires another Magic Missile at any target that presents itself.
Damage 5d4 + 5 ⇒ (1, 2, 3, 2, 1) + 5 = 14

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Aiding Again!: 1d20 ⇒ 12
Mathais points out exactly how Jagger needs to drive the cart, after all, Mathais is a great backseat driver. :D

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If they're within 190' and Jarlgur can see them, he'll fire at the other cart, preferably the woman. If not, he'll cast guidance.
Possible Longbow Attack: 1d20 + 9 ⇒ (6) + 9 = 15
Possible Damage: 1d8 + 4 ⇒ (6) + 4 = 10
Possible Iterative Longbow Attack: 1d20 + 4 ⇒ (2) + 4 = 6
Possible Damage: 1d8 + 4 ⇒ (4) + 4 = 8

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Don't forget the +2 for aiding :D

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Maaarv will draw and load his light crossbow, hoping that he'll be able to get a clear shot soon.
(2 move actions)

GM Bold Strider |

The cart violently shakes as Jaggar mistimes the steering at a sharp turn, slamming the cart into the wall. The group is jostled severely causing everyone to slam into each other.
Non-lethal Damage to Everyone: 1d6 ⇒ 2
Alie takes out another man with a blast of magic missiles. He crumples to the floor of the mine cart in front of them as Jarlgur lets loose two arrows. Sadly, his aim isn't as good as the spell's and they clatter against the rocky corridor.
2d20 ⇒ (5, 16) = 21
Their cart speeds up as the woman throws the body of the man onto the tracks. The body tumbles out, splaying across the rails. She quickly drops down behind the steel wall of the cart to avoid any magical force damage.
@Jaggar: Three Ride checks; Brace for the body, Stabilize for speed, Speed up the cart
Initiative - Cart: Speed=80'/rd.; 500' travelled
1st: Jaggar - Driving Cart @ 2 NL; Mathais @ 2 NL;
2nd: Woman @ 7 (On Cart); Guards (#1-3 @ 4 - On Cart) [90'/rd.] [720']
3rd: Jarlgur @ 2 NL; Alie @ 2 NL; Maaarv @ 2 NL