The tomb of the unknown ruler

[Prof. Templeton]

Professor Templeton’s great-great-grandfather, Aloysius Templeton, was a well known explorer and relic collector. One of his collected histories told of a site near Ugarit. There was located an ancient temple and beneath a burial chamber. The identity of the ruler entombed within was lost to the ages until a tome was discovered that shed light on the question. All that was previously known of the long dead leader was that he was one of the three sons of his predecessor, Zimilkar.

The tome had been translated like this:

Zimilkar had three sons, Ammit, Biranu and Canthar. The aged king knew his time was almost due but could not decide among which of his three sons to leave the responsibility of leading the people. He sent each out with many men to mine an equal number of cubic stones of the same size used to construct the city’s central plaza and whoever was first to build a square around the plaza, it also being a square, would be the next ruler.

Biranu returned to the city to find Ammit had gotten there first and was already building a square. A clever man and not one to be outdone by his sibling, Biranu set his men to builduing his square around his brothers. When Canthar returned last and saw what his brothers were doing, he also proceeded to have his square built around Biranu’s.

It happened that all three sons finished their projects on the same day so that when the old king came to inspect he could not tell who had finished first. He noted that both Ammit and Biranu had used all of their stone cubes, but Canthar had four stones left over and on this basis he was disqualified. Among the remaining two sons a contest was devised to determine the successor.

They would have a race upon the central plaza. Biranu, being more athletic and boastful decided to give his brother an advantage. He would allow Ammit to start from the Southern corner of the plaza while he would start from the West and both would finish in the East. Biranu’s path, however, would not be straight. It was to join with Ammit’s path one quarter of the way from the finish along the distance Ammit was to run.

Both men were given the signal to start and they ran as fast as they could. When Biranu reached the point where his path and Ammit’s joined together he saw Ammit was ahead of him by a distance equal to one twentieth of the the total distance Biranu had to cover . It was a well ran race and a close finish.

The rest of the tome is lost to decay, but the winner was still found with the information given and indeed the site of the central plaza was also found by determining the original size and comparing it with the ruins that still remained.

Is the entombed ruler Ammit or Biranu or perhaps neither?

[CaptainEd]

If the side of the plaza square is 4, Biranu has to run the hypotenuse of a 3-4-5 right triangle in order to join Ammit's course. So Biranu's total course is 6 units, while Ammit's is 4. When Biranu arrives at Ammit's path, Ammit is 6/20 (=.3)ahead of him. So Ammit has run 3.3 in the time it took Biranu to run 5. Biranu runs the remaining 1 distance at a rate of 5, finishing in 0.2 units of time. Ammit runs the remaining 0.7 distance at a rate of 3.3, finishing in 0.21 units of time. Biranu wins

But I clearly can't get into the minds of the Ugarits--

I interpreted the phrase "...to mine an equal number of cubic stones" to mean that A, B, and C are to have the same number of stones. If so, I'm stumped about the following.

it sounds like C's square was built around B's square was built around A's square, and yet C used fewer stones than B, and B and A used the same number of stones.

Do you feel that each person built a complete square? Or do you think each person built some walls, and that the combined assemblage exhibited 3 interlocking squares? Or something else?

[curr3nt]

Let x = length of the original square.

a = new length of square after Ammit is done

b = new length of square after Biranu is done

c = new length of square after Canther is done

a^2 - x^2 = b^2 - a^2 = c^2 - b^2 + 4

I think this will give each son the same number of stones and fit the puzzle... not sure what values would work yet.

[curr3nt]

Original square was 34 cubics by 34 cubics for a total of 1156 stones.

Ammit added 1344 stones to make it 2500 stones. (50 by 50)

Biranu added 1344 stones to that to make a total of 3844 stones. (62 by 62)

Canthar added 1340 stones for a total of 5184 with 4 stones left over (72 by 72)

Not sure if other values would also work.

[Prof. Templeton]

If the side of the plaza square is 4, Biranu has to run the hypotenuse of a 3-4-5 right triangle in order to join Ammit's course. So Biranu's total course is 6 units, while Ammit's is 4. When Biranu arrives at Ammit's path, Ammit is 6/20 (=.3)ahead of him. So Ammit has run 3.3 in the time it took Biranu to run 5. Biranu runs the remaining 1 distance at a rate of 5, finishing in 0.2 units of time. Ammit runs the remaining 0.7 distance at a rate of 3.3, finishing in 0.21 units of time. Biranu wins

Original square was 34 cubics by 34 cubics for a total of 1156 stones.

Ammit added 1344 stones to make it 2500 stones. (50 by 50)

Biranu added 1344 stones to that to make a total of 3844 stones. (62 by 62)

Canthar added 1340 stones for a total of 5184 with 4 stones left over (72 by 72)

Not sure if other values would also work.

You are both correct. Well done! When I was writing this I realized that even without the first half the winner of the race could be found as the Captain did. I was going to change the distance of where the paths met from one fourth to a number of stones to make it harder, but then forgot to do so.