## Measuring Hobbiton

An elf and a hobbit measured the length of the latter’s piece of land. The hobbit did so by taking 54 cm long steps.After him, the elf measured again by taking 72 cm long steps. After that there were 61 footprints in the mud, but sometimes one footprint corresponded to two footsteps, one from the hobbit and one from the elf. What was the length of the piece of land?

hex| PUZZLE MASTER | Profile December 11th, 2009 - 6:20 amI assume the N=61 footprints include the starting footprints.

L is the land’s length.

Nh and Ne are the hobbit and elf’s footprints respectively:

L = 54 x (Nh – 1)

L = 72 x (Ne – 1)

Overlapping footprints: L = 12×18 x (No – 1) as 12×18 is smallest number that can be divided by both 54 (3 x 18) and 72 (4 x 18).

(N – 1) = (Ne – 1) + (Nh – 1) – (No – 1)

Hence L = 10 x 12 x 18 = 2160 cm = 21.6 m

Nh = 41

Ne = 31

Common footprints = 11

joe| Profile December 11th, 2009 - 6:43 amEvery 3 steps that the elf takes coincides with the fourth step that the hobbit takes.

(3 x 72cm = 4 x 54 cm)

so at this point there would be 6 footprints visible (the last one coinciding between the two)

If there are 61 prints in the mud then up to the 60th footprint there would be a length of 20 of the hobbits (or 15 of the elf´s) either way a length of 1080cm. The 61st footfrint must be that of the hobbit, the 54 cm, so the length of the land is 1080+54 = 1134cm

suineg| PUZZLE MASTER | Profile December 11th, 2009 - 10:13 amI have some doubts regarding the problem:

partial footprints counts? if elf were after the hobbit and measure exactly at the same path, the only footprints that remains re the one of the Elf because there were bigger or you can still see the hoobit footprints?, if so, when one footprint correponded to two footstep is when both steps begin in the same point like the 216 cm where both man start a step in the measuring journey, because they are always mixed right?

cool puzzle.

Shawn| PUZZLE GRANDMASTER | Profile December 11th, 2009 - 10:54 amI will assume that the first footprint is the starting point for the munchkins, which is a shared footprint corresponding to distance = zero.

In that case, the tromp ended in a shared footprint as well, at 2,160cm.

Because the step distances are a ratio of 3:4, every 4th step by the Hobbit is coincidental with every 3rd step of the Elf. In our equation, we have to account for this by subtracting out one of the shared footsteps, or (as I did) 1/2 of each shared footstep. We also have to add in the 1st footstep at distance = zero

H = # of Hobbit steps

E = # of Elf steps

X = total length of land

61 = total number of footprints

54*H = X

72*H = X

1 + [H - (H/4)*(1/2)] + [E - (E/3)*(1/2)] = 61

Substitution and solution shows:

H = 40 steps

E = 30 steps

X = 2,160cm

mimi112| Profile December 11th, 2009 - 11:56 am2160 cm

ahsergio| Profile December 11th, 2009 - 12:59 pm2196m

And the last footprnt belongs to the hobbit!

Falwan| Profile December 11th, 2009 - 11:04 pmMany thanks Dr. RK for asking about my absence.

Had to have a back disk operation,and I feel O.K. now.

For this puzzle I think no need for complex equations and calculations, because the guy with shorter steps is going to take the 61 steps…

so, 61 × 54 = 3294 cm = 32.94 meters

Falwan| Profile December 11th, 2009 - 11:05 pmdisk = disc…spelling mistake again!!

LadyInsomnia| Profile December 12th, 2009 - 1:07 amThe length of the piece of land is 2160 cm.

3*72 = 4*54

elf-steps:

0 3 6 9 12 15 18 21 24 27 30

hobbit steps:

0 4 8 12 16 20 24 28 32 36 40

step “0″ still leaves a print

31 + 41 = 72 possible footprints

72 – 11 matching footprints = 61 visible footprints

30 elf-steps = 30*72cm = 2160cm

or

40 hobbit-steps = 40*54cm = 2160cm

abcbcdcdef| Profile December 12th, 2009 - 10:32 am2160cm?

Falwan| Profile December 12th, 2009 - 11:56 pmOops! I change my mind:

54cm long step:

40 steps × 54 cm = 2160 cm

72cm long step:

72 steps × 30 steps = 2160 cm

That is 60 steps only.

The 60th step will come from the guy with shorter steps;

so, 2160 + 54 = 2214 = 22.14 meters

Mashplum| PUZZLE MASTER | Profile December 13th, 2009 - 7:07 pmThe first footprint corresponds to zero and was made by both creatures. Every 216 cm (the LCM) there will be 4 new hobbit prints and 3 elven – one of which will overlap – leaving only 6 visible. To contain 61 prints, the land must be 2160 cm or 21.6 m long.

Mashplum| PUZZLE MASTER | Profile December 13th, 2009 - 7:14 pmMy solution presumes no difference between left and right feet. If we take that into consideration, it would be 13 new distinct prints every 432 cm. Since 60 is not divisible by 13, I will assume this is not how the problem is to be approached, but the result would be 432×4 + 216 + 54 = 1998 or 19.98m

Bobo The Bear| PUZZLE MASTER | Profile December 14th, 2009 - 5:48 pmSolution: The plot of land is exactly 2160 cm long.

The overlapping footprints occur every 216 cm, because 216 is the least common multiple of 54 and 72. Every 216 cm, the hobbit walks 4 paces and the elf walks 3 paces. For every 216 cm, what you see on the ground is 3 hobbit prints, 2 elf prints, and one print where they overlap. So every set of 6 prints corresponds to 216 cm paced off.

If they both started at the inside edge of the plot of land, then print #1 is an overlap, as are prints 7, 13, 19, 25, and so on to print 61. This means that they paced off 216 cm a total of 10 times, for a distance of 2160 cm.

Disclaimer: There are a few assumptions I chose to make, because they led to an exact answer, rather than an indefinite range.

The first is that they started inside the plot of land. If they started outside, then the overlaps are at prints 6, 12, 18, and so on to print 60. The last print would be from the hobbit, and the plot of land is somewhere between 2214 cm and 2232 cm long.

The second is that, if one’s left foot and the other’s right foot correspond to the same distance, it would be considered an overlap, rather than two prints side by side. If the latter is the case, then there would be 13 prints every 432 cm instead. If they both started inside the plot of land and both started off with the same foot, the overlaps would be at prints 1, 14, 27, 40, and 53, covering the first 1728 cm. There would be 5 additional hobbit prints and 3 additional elf prints, making the total distance somewhere between 2318 cm and 2336 cm in length.

Sorry that the disclaimer is longer than the original solution, but sometimes that’s the price you pay for a complete answer.

Bobo The Bear| PUZZLE MASTER | Profile December 14th, 2009 - 5:52 pmI made an error in my last calculations: In my final disclaimer, the total distance would be between 1998 cm and 2016 cm (not 2318 cm and 2336 cm).

I really have to do a better job proofing my own work before I submit. I think I’m starting to lose it in my old age.

ohjirochan| Profile December 14th, 2009 - 7:41 pmThe length of the hobbit’s land is 2160 cm.

You can find a short pattern with a simple chart by measuring step by step and find that for every 4 steps the hobbit takes the elf 3 steps. That final step corresponds to two foot prints. Thus leaving 6 foot prints on the ground. This pattern repeats until a total of 60 footprints are made afterwhich the hobbit took 40 steps and the elf took only 30 both of which add up to 2160 cm.

And here’s the rub, where did footprint 61 come from………. the starting point.

RogHyde| Profile December 15th, 2009 - 2:12 pm2214cm

For each 216cm the Hobbit would take 4 steps (216/54=4) and the Elf would take 3 steps (216/72=4) but one of these steps would coincide so only 6 footsteps would appear for each 216cm.

After 2160cm there would be 60 steps so after the next step (the Hobbit’s) the total would be 2160 + 54 = 2214cm.

xpress84| Profile December 16th, 2009 - 1:48 amapprox. 2160cm

ahsergio| Profile December 16th, 2009 - 7:45 amyey, answer not unveiled yet, presenting this problem to a friend i realized my mistake, the length of hobbits land is 2212cm or 22,12m. not 2196 as i answered before

RK| Founder | Profile December 16th, 2009 - 9:51 pmVery good, the answer Bilbao had was 2160. Unfortunately due to some things that have come up, he won’t be able to visit the site again until sometime next week.

However, several of our longtime regular readers have excellent explanations you can check out above.

ahsergio| Profile December 18th, 2009 - 6:14 amtwo wrong answers, forgot that step 0 would leave a print =(

allways good to add something new though

RogHyde| Profile January 17th, 2010 - 3:38 pmI do not log on to this site very often so have only just read some of the posts.

I had presumed in my earlier answer that the hobbit and elf would start with their feet outside the land to measure each pace from the front of one footprint to the front of the next.

But if their first footprint was inside the plot of land then they were measuring from the rear of their first footprint to the rear of the next one.

Therefore after the hobbit had made 41 steps, including the first step at zero (and the elf had made 31 steps including the first step at zero, only 20 of which would not coincide with the hobbit’s), the measurement to the back of their final footprint would be 2160, so if this footprint (number 61) is to be included, then the total would be anything between (2160 + the length of the last footprint) and 2214 which would be where the next hobbit footprint (which would be number 62 so is not within the plot of land) would be.