perplexus.info :: Just Math : How Mean Is It?

How Mean Is It? (Posted on 2023-10-07)

The teacher has recently set a class the following puzzle: "Write down four positive whole numbers less than 100 and then write down their average: choose the original numbers so that overall the five numbers written down use each of the digits 1 to 9 exactly once."

Three students in the class were successful. Heinrich came up with 9, 15, 36 and 48, with average 27. Felix was more cunning – his average was not a whole number – it involved a mixed fraction in its simplest form, like 27¾. Delphine was devious, too, and her average, which was lower than Felix's, was not a whole number – it involved decimals, like 16.25.

What averages did Felix and Delphine write?

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Computer solution

Comment 2 of 2 |

Felix's average: 25 3/4

Delphine's average: 24.75 (which was lower than Felix's)

Program Output:

Felix

1,7,9,86 with average 25 3/4

1,6,9,87 with average 25 3/4

1,6,7,89 with average 25 3/4

6,7,9,81 with average 25 3/4

Delphine

1,6,9,83 with average 24.75

1,3,9,86 with average 24.75

1,3,6,89 with average 24.75

3,6,9,81 with average 24.75

7,9,24,86 with average 31.5

6,9,24,87 with average 31.5

6,7,24,89 with average 31.5

7,9,26,84 with average 31.5

4,9,26,87 with average 31.5

4,7,26,89 with average 31.5

6,9,27,84 with average 31.5

4,9,27,86 with average 31.5

4,6,27,89 with average 31.5

6,7,29,84 with average 31.5

4,7,29,86 with average 31.5

4,6,29,87 with average 31.5

8,9,42,67 with average 31.5

7,9,42,68 with average 31.5

7,8,42,69 with average 31.5

8,9,47,62 with average 31.5

2,9,47,68 with average 31.5

2,8,47,69 with average 31.5

7,9,48,62 with average 31.5

2,9,48,67 with average 31.5

2,7,48,69 with average 31.5

7,8,49,62 with average 31.5

2,8,49,67 with average 31.5

2,7,49,68 with average 31.5

----------------------------------

import math

from itertools import permutations

print('Felix')

for p in permutations('123456789'):

if int(p[7]) >= int(p[8]):

continue

if math.gcd(int(p[8]),int(p[7])) != 1:

continue

a = int(p[1])

b = 10*int(p[2])+int(p[3])

c = int(p[4])

d = int(p[5])

if a>c or a>d or c>d:

continue

avg = 10*int(p[0])+int(p[6]) + int(p[7])/int(p[8])

if avg == (a+b+c+d)/4:

print('{},{},{},{} with average {} {}/{}'.format(a,c,d,b,10*int(p[0])+int(p[6]),int(p[7]),int(p[8]) ))

print('Delphine')

for p in permutations('123456789'):

a = int(p[1])

b = 10*int(p[2])+int(p[3])

c = int(p[4])

d = int(p[5])

if a>c or a>d or c>d:

continue

avg = 10*int(p[0])+int(p[6]) + int(p[7])/10 +int(p[8])/100

if avg == (a+b+c+d)/4:

print('{},{},{},{} with average {}.{}{}'.format(a,c,d,b,10*int(p[0])+int(p[6]),int(p[7]),int(p[8]) ))

for p in permutations('123456789'):

a = 10*int(p[0])+int(p[1])

b = 10*int(p[2])+int(p[3])

c = int(p[4])

d = int(p[5])

if a>b or c>d:

continue

avg = 10*int(p[6]) + int(p[7]) + int(p[8])/10

if avg == (a+b+c+d)/4:

print('{},{},{},{} with average {}.{}'.format(c,d,a,b,10*int(p[6])+int(p[7]), int(p[8]) ))

Posted by Larry on 2023-10-07 11:43:15

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