Answer to the Problem of the Week
for the week of February 7, 2005

Eight Integers and Eight Clues

I have eight integers in my list. Use the clues to find my numbers.

  1. The range (statistical sense) of the numbers in my list is 93.
  2. One of my numbers is an abundant number between 21 and 25.
  3. One of the numbers in my list is a perfect cube less than 10.
  4. The arithmetic mean of the numbers in my list is 40.
  5. The mode of the list is the sum of the first two perfect numbers.
  6. The median of the numbers in my list is 29.5.
  7. The largest number in my list is the smallest 3-digit palindrome.
  8. The sum of the digits of one of the numbers is its square root.

What are the eight numbers?

 

Solution to the Problem:

The numbers are 8, 13, 24, 25, 34, 34, 81, 101.

  • Clue 7. The largest number is the smallest 3 digit palindrome: 101
  • Clue 1. The range, maximum - minimum, is 101 - 8 =93, so the smallest number must be 8.
  • Clue 3. A cube less than 10: that would have to be 8=2^3 or 1=1^3. This clue is not really needed because of Clue 1.
  • Clue 4. The arithmetic mean of 8 numbers is 40, so the sum of the numbers is 8*40=320.
  • Clue 2. The first few abundant numbers (numbers such that the sum of the divisors less than the number is greater than the number) are 12, 18, 20, 24, 30, ... hence the abundant number between 21 and 25 would have to be 24.
  • Clue 5. The mode of the list is the sum of the first two perfect numbers, numbers such that the sum of the proper divisors equal the ,number. The first two are 6 and 28, hence the mode is 34. Since a mode would not be defined unless at least two numbers in the list are the same, we assume two number 34s.
  • Clue 6. The median is 29.5=mean of two successive entries, one of which is the maximum of the lower four numbers and one of which is the minimum of the upper four numbers, e.g., 29 and 30, or 28 and 31, or 27 and 32, or 26 and 33, or 25 and 34. Aha! We've got it: 25 and 34.
  • Clue 8. Looking at a table of squares, the only one under 121 such that the sum of its digits is the root is 81. . . "Summing up"; from least to greatest we have:
    8, x, 24, 25, 34, 34, 81, 101. x is a slack variable, chosen so that the sum of the numbers is 320, that is x=13. There they are!

Correctly solved by:

1. Arin Smith Winchester, Virginia
2. John Funk Ventura, California
3. Sara Christopher Columbus, Georgia
4. Kirstine Wynn St. Olaf's College,
Northfield, Minnesota
5. David & Judy Dixon Bennettsville, South Carolina
6. Nathan Seifert Harrisonburg, Virginia
7. Sara Major Winchester, Virginia
8. Jeffrey Gaither Winchester, Virginia
9. Crystal Church Columbus. Georgia
10. Sharina Broughton Old Dominion University,
Norfolk, Virginia
11. Dave Smith Toledo, Ohio
12. Ben Swartz Winchester, Virginia
13. Emily Auerbach Columbus, Georgia
14. Leslee Champion Columbus, Georgia
15. Sam Coffin Columbus, Georgia
16. Laura Rickman Columbus, Georgia
17. Tristan Collins Winchester, Virginia
18. Paige Janke Columbus, Georgia
19. Tarpley Ashworth Harrisonburg, Virginia
20. Camron S. Columbus, Georgia
21. Sarah Jane Columbus, Georgia
22. Okechi Egekwu Harrisonburg, Virginia
23. Justin Crabill Harrisonburg, Virginia
24. Jonathan "DenRyskeHAxxEr" Jansson,
    Bahadir "Vladimir" Güngör,
   Ako "DimitRoV" Saleh 		Tullängsskolan, Örebro, Sweden
25. Wajih Ansari Harrisonburg Virginia
26. Arsalan Heydarian Harrisonburg Virginia
27. Rebecca Crawford Columbus, Georgia
28. Jeanette Crawford Columbus, Georgia
29. Libba Richardson Columbus, Georgia
30. Tom Robb Winchester, Virginia
31. Amanda Auerbach Columbus, Georgia
32. Trey Mason Winchester, Virginia
33. Mikael Holmquist Tullängsskolan, Sweden
34. Johnas Eklof Tullängsskolan, Örebro, Sweden
35. Linus Oskarsson Tullängen, Sweden
36. Erik Hagberg Tullängsskolan, Sweden
37. Jonas Melin Tullängsskolan, Sweden
38. Weimers Winners Lordstown, Ohio
39. Gilbert Melanson ----------
40. Cameron Burkholder Winchester, Virginia
41. Misty Carlisle Winchester, Virginia