# A 'Game of Thrones' maths puzzle

Four prisoners escape from the dungeons at King's Landing and head for the Wall; a distance of 700 leagues. The Maester travels 1 league in the first day, then each subsequent day, doubles the distance he travelled the previous day. The Whore travels 50 leagues each day. The Knight travels 80 leagues each day, and 20 leagues each night. The Squire travels 350 leagues the first day, then each subsequent day, travels half the distance he travelled the previous day.

Two days after the prisoners escape, the Mountain is sent out to hunt them down and kill them. He travels 100 leagues each day. The Maester has a 5% chance of talking the Mountain out of killing him. The Whore has a 20% chance of seducing the Mountain and escaping death. The Knight has a 45% chance of defeating the Mountain in combat. The Squire has a 70% chance of evading the Mountain. All prisoners risk being attacked and killed by wolves; a 5% chance each day, and a 25% chance each night. If any prisoner reaches the Wall they 'take the black' and live out their days in the Night's Watch.

Assuming all prisoners have enough food, water and clothing, which of them is guaranteed to die before they reach the Wall, and how will they die?

• Does is make a difference if the prisoner is travelling at night for the chance of wolves killing them? The knight is the only one who travells during night, so does the killed by wolveschance each night only counts for him?
– tyui
Mar 23, 2020 at 11:02
• @tyui No, wolves have a chance of killing you whether you're travelling at night or not. Mar 23, 2020 at 11:10
• Can the Maester, the Whore and the Squire travel at night? Mar 23, 2020 at 11:17
• @trolley813 Nope. Mar 23, 2020 at 11:57
• What if someone faces the mountain more than once? Does he stop at the wall and wait for them? Do they get a second chance? If the knight defeats him in combat, does he die? Mar 23, 2020 at 21:07

The one who is guaranteed to die before they reach the Wall is:

The Squire

Because:

The total distance is 700 leagues, and he travels 350 on the first day. Which is half the distance. He then travels half as much the next day, which is also half of the remaining distance.
If he travels half of the remaining distance, then he will never reach, but get infinitesimally closer, until he eventually succumbs to either the Wolves, The Mountain, or old age. Whilst there is a very small chance that the wolves/mountain won't kill him, Old Age is guaranteed to get him eventually.
The other people will all eventually make it, if they avoid the Mountain/wolves, but they would never be guaranteed to die, they only have a chance of dying, and a chance of getting to the Wall.

• Most of what you said is correct, except for one small detail... Mar 23, 2020 at 10:13
• Still the squires has an infinitysmall chance that he won't be killed by the wolves (or the mountain), thus not guaranteed to die. Although, I do agree it has the biggest change of dying.
– tyui
Mar 23, 2020 at 11:42
• I want to comment more on this answer, but comments don't allow spoilers... :-( Mar 23, 2020 at 11:59
• Mathematically speaking, he will be almost surely killed. Mar 23, 2020 at 12:02
• Made an edit for their eventual death condition Mar 23, 2020 at 14:08

# Travel time

assuming you reach the wall if you have traveled at least 699 leagues(to be generous to the squire) and there were no mountain or wolves. the prisoners would have to travel for a number of days:
10 days for the Maester. $$f(d) = 2 ^ d$$
14 days for the Whore. $$f(d) = 50$$
7 days for the Knight. $$f(d) = 100$$
10 days for the Squire. $$f(d) = 350 * 2 ^ -d$$
f(d) is the distance the prisoner traveld after d days.
the above days is the lowest number of days where $$integral(f(d)) > 699$$

# The real danger

every day there is a 95 % chance that the prisoner is left alone by the wolves. and every night there is a 75 % chance that the prisoner is left alone by the wolves. so every full day there is a 71.25% chance (.95 * .75) that the prisoner is not attacked by wolves. for each prisoner the chance to take the black reduces for every night the wolves might attack and each prisoner can be confronted by the mountain (the knight travels at the same speed as the mountain and has a two day head start so the knight can not be confronted).
so the chance to survive the trip is $$0.7125 ^ d * c$$ where d is the number of days and c is the chance of surviving the confrontation with the mountain. as the number of days increases. this function rapidly reduces to 0%. because as the days go on the chance that no wolves have attacked goes to zero.

# Taking the black?

the Maester has 0.169% chance to survive. the Whore has a 0.174% chance to survive. the Knight has a c of 1 because the mountain can not overtake him and the knight has the lowest amount of days yielding him a 9.3% chance to survive. the Squire has a good chance of surviving his encounter with the mountain and has a 2.3 % chance of survival(not counting the squire being practicly paralized after 10 days and therefor unable to cross the last league to take the black)

# Conclusion

Almost surely all prisoners will be gobbled up by wolves, especially the squire.
there is a slim chance that the prisoners will get killed by the mountain.
And the knight might make it, if the wolves are not particularly hungry when he escapes.

• According to your formulas, in the first day the maester travels 2 leagues, and the squire travels 700 leagues. Mar 24, 2020 at 1:19
• I guess i missed the part that i start counting days at 0 Mar 24, 2020 at 8:31
• And i also missed a minus sign in the formula of the squire Mar 24, 2020 at 8:33