"My daughter (Y5) sometimes uses a trial and error approach to solve such problems. A random (easy) trial and error process is to use 1hr each way on level ground, 2hrs uphill and 1hr downhill. It then becomes obvious that all we need to do is increase the level ground time to 1.5hrs each way in order for the total time to be 6hrs. We then got the following for the distance:
1.5hrs x 4 km/h + 2hrs x 3km/h + 1hr x 6 km/h + 1.5x4 km/h =24 km
(Level + uphill + downhill + level)
Thank you! This was wonderful and much enjoyed!
"
Beth, White Bear Lake, MN
Wednesday, December 4, 2024
"Let x = time on level ground, y = time traveling uphill, and z = time traveling downhill.
He walked for 6 hours.
x+y+z=6
He traveled the same distance uphill and downhill.
3y=6z
y=2z
x+y+z=6
x+2z+z=6
x+3z=6
x=6-3z
He walked 4 km/h on level ground, 3 km/h uphill, and 6 km/h downhill.
We want the total distance
=4x+3y+6z
=4(6-3z)+3(2z)+6z
=24-12z+6z+6z
=24
Prancer walked a total of 24 km.
"
Rick, US
Saturday, December 7, 2024
"Let x represent the one-way distance on level ground and y represent the one-way distance of the hill. Then x/4 represents the time he walked on level ground, y/3 is the time to walk uphill, and y/6 is the time to walk downhill. This results in the following equation.
x/4+y/3+y/6+x/4 = 6
combining and reducing fractions (in high-school math class, I would have had to separate this into multiple steps. Arg!) yields:
x/2+y/2 = 6 or
x + y = 12
Since x and y represent one-way distances, being 12 km, round trip distance is 24 km."
Eorl, Sydney, Australia
Monday, December 9, 2024
"I had to relisten but as we are not given anything to distinguish between the flat and hill, the answer must be the same regardless.
So pretend its all flat - a simple 6h * 4km/h, reveals a total distance of 24km (or 12km there and 12km back again).
This can be confirmed by pretending its all hill - 12km up @ 3km/h takes 4h, 12km down at 6km/h takes 2h, which correctly adds to 6h.
"
Rick, US
Saturday, December 28, 2024
"I have come up with an alternate solution to the December 2024 puzzle.
Reading the puzzle, there does not appear to be sufficient information to determine the flat versus incline distances. With this in mind, it might be interesting to determine the average pace for the inclined path.
From the engineering part of me:
Assume that the incline path is six km long. Then Prancer needs two hours to stroll uphill and one hour to stroll downhill, for a total of three hours to traverse 12 km, for an average pace of four km per hour.
From the mathematical part of me:
Assume that the inclined portion of the trip is x km. then it takes Prancer x/3 hours to walk uphill and x/6 hours to walk downhill. Adding these two times and reducing results int x/2 hours to stroll a distance of 2x km. 2x divided by x/2 results in 4 km per hour.
Since the average pace is 4 km per hour regardless of the terrain, simply multiplying the average pace and the number of hours (6) result in the distance travelled (24 km).
"
Steve Walker, BlueSky
Monday, January 13, 2025
2025 is a square number,
but did you also spot that 2025 is the product of a prime number, a square number & a cube number.
2025 = 3 × 25 × 27
What is the next year that will also be the product of a prime number, a square number and a cube number?
#OCRMathsPuzzle #UKMathsChat #RecreationalMaths
Kevin, Australia
Sunday, December 1, 2024
"Let a be the time walking uphill then a/2 is the time walking downhill.
Let b be the time walking on the level.
So a +0.5a + b = 6 and 1,5a = 6 – b
Tot dist = 3a + 6a/2 + 4b = 6a + 4b = 4(6 – b) + 4b = 24km "
Tzvetanka,
Monday, December 2, 2024
"My daughter (Y5) sometimes uses a trial and error approach to solve such problems. A random (easy) trial and error process is to use 1hr each way on level ground, 2hrs uphill and 1hr downhill. It then becomes obvious that all we need to do is increase the level ground time to 1.5hrs each way in order for the total time to be 6hrs. We then got the following for the distance:
1.5hrs x 4 km/h + 2hrs x 3km/h + 1hr x 6 km/h + 1.5x4 km/h =24 km
(Level + uphill + downhill + level)
Thank you! This was wonderful and much enjoyed! "
Beth, White Bear Lake, MN
Wednesday, December 4, 2024
"Let x = time on level ground, y = time traveling uphill, and z = time traveling downhill.
He walked for 6 hours.
x+y+z=6
He traveled the same distance uphill and downhill.
3y=6z
y=2z
x+y+z=6
x+2z+z=6
x+3z=6
x=6-3z
He walked 4 km/h on level ground, 3 km/h uphill, and 6 km/h downhill.
We want the total distance
=4x+3y+6z
=4(6-3z)+3(2z)+6z
=24-12z+6z+6z
=24
Prancer walked a total of 24 km. "
Rick, US
Saturday, December 7, 2024
"Let x represent the one-way distance on level ground and y represent the one-way distance of the hill. Then x/4 represents the time he walked on level ground, y/3 is the time to walk uphill, and y/6 is the time to walk downhill. This results in the following equation.
x/4+y/3+y/6+x/4 = 6
combining and reducing fractions (in high-school math class, I would have had to separate this into multiple steps. Arg!) yields:
x/2+y/2 = 6 or
x + y = 12
Since x and y represent one-way distances, being 12 km, round trip distance is 24 km."
Eorl, Sydney, Australia
Monday, December 9, 2024
"I had to relisten but as we are not given anything to distinguish between the flat and hill, the answer must be the same regardless.
So pretend its all flat - a simple 6h * 4km/h, reveals a total distance of 24km (or 12km there and 12km back again).
This can be confirmed by pretending its all hill - 12km up @ 3km/h takes 4h, 12km down at 6km/h takes 2h, which correctly adds to 6h. "
Rick, US
Saturday, December 28, 2024
"I have come up with an alternate solution to the December 2024 puzzle.
Reading the puzzle, there does not appear to be sufficient information to determine the flat versus incline distances. With this in mind, it might be interesting to determine the average pace for the inclined path.
From the engineering part of me:
Assume that the incline path is six km long. Then Prancer needs two hours to stroll uphill and one hour to stroll downhill, for a total of three hours to traverse 12 km, for an average pace of four km per hour.
From the mathematical part of me:
Assume that the inclined portion of the trip is x km. then it takes Prancer x/3 hours to walk uphill and x/6 hours to walk downhill. Adding these two times and reducing results int x/2 hours to stroll a distance of 2x km. 2x divided by x/2 results in 4 km per hour.
Since the average pace is 4 km per hour regardless of the terrain, simply multiplying the average pace and the number of hours (6) result in the distance travelled (24 km). "
Steve Walker, BlueSky
Monday, January 13, 2025