Add a fifth number to the set {3,7,9,10} so that the mean is equal to the median of the set.
Show whether or not it is possible to create a set of numbers where the same problem yields more solutions than the above.
If we are to make the 7 the median and mean, then we need to make the total 35. The total is already 29 so we need to add a 6.
If we are to make the 9 the median and mean, then we need to make the total 45. The total is already 29 so we need to add a 16.
This allows all integral members of the set. Ady has shown a way of making the new member be the mean and median, by using a non-integer.
In all cases, the maximum is three possibilities: either of the two existing central numbers or the newly added member.
Some cases where a set of four can have three numbers serve as answer and all be integers are:
original numbers possible add-ons
3 5 7 9 1 11 6
3 6 8 11 2 12 7
3 6 9 10 2 17 7
3 8 10 15 4 14 9
3 8 11 14 4 19 9
3 8 12 13 4 24 9
3 9 11 17 5 15 10
3 9 12 16 5 20 10
3 9 12 20 1 16 11
3 9 13 15 5 25 10
3 9 13 19 1 21 11
3 9 14 18 1 26 11
3 9 15 17 1 31 11
3 10 12 19 6 16 11
3 10 13 18 6 21 11
3 10 13 22 2 17 12
3 10 14 17 6 26 11
3 10 14 21 2 22 12
3 10 15 16 6 31 11
3 10 15 20 2 27 12
3 10 16 19 2 32 12
3 10 17 18 2 37 12
3 11 13 21 7 17 12
3 11 14 20 7 22 12
3 11 15 19 7 27 12
3 11 16 18 7 32 12
3 12 14 23 8 18 13
3 12 15 22 8 23 13
3 12 15 26 4 19 14
3 12 16 21 8 28 13
3 12 16 25 4 24 14
3 12 17 20 8 33 13
3 12 17 24 4 29 14
3 12 18 19 8 38 13
3 12 18 23 4 34 14
3 12 19 22 4 39 14
3 12 20 21 4 44 14
3 13 15 25 9 19 14
3 13 16 28 5 20 15
3 13 17 23 9 29 14
3 13 17 27 5 25 15
3 13 17 31 1 21 16
3 13 18 22 9 34 14
3 13 18 26 5 30 15
3 13 18 30 1 26 16
from
CLS
FOR a = 3 TO 10
FOR b = a + 1 TO 20
FOR c = b + 1 TO 40
FOR d = c + 1 TO 60
sum = a + b + c + d
avg = sum / 4
IF avg = INT(avg) THEN
IF avg > b AND avg < c THEN
sum1 = 5 * b: sum2 = 5 * c
new1 = sum1 - sum: new2 = sum2 - sum
IF new1 <> a AND new1 <> b AND new1 > 0 AND new2 <> c AND new2 <> d THEN
PRINT a; b; c; d, new1; new2; avg
lct = lct + 1
END IF
END IF
END IF
IF lct > 44 THEN lct = 0: PRINT : STOP
NEXT
NEXT
NEXT
NEXT
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Posted by Charlie
on 2014-02-20 16:07:36 |
solution
