All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Logic
Number Pyramid (Posted on 2004-08-20) Difficulty: 2 of 5
A "Number Pyramid" is composed of ten different numbers ( usually 0 - 9 ) with four rows. ONE EXAMPLE of a Number Pyramid is as follows:

      0

    1   2

  3  4   5

6  7   8   9

Given the clues below, determine the composition of a new Number Pyramid.

1) The sum of the two numbers in the second row is 11.

2) The sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10.

3) The rightmost numbers in the four rows must sum to 18.

4) The middle number in the third row minus the leftmost number in the second row should equal 4.

5) Subtracting the top number in the Pyramid from the rightmost number in the second row leaves 4.

6) In the bottom row, the leftmost number is greater than the second number from the left, while the rightmost number is greater than the second number from the right.

For bonus marks, which of the above clues (if any), are not necessary in order to construct the pyramid?

  Submitted by Jane Doe    
Rating: 3.5556 (9 votes)
Solution: (Hide)
By clue 1, the sum of the two numbers in the second row equals 11, so that the leftmost number must be 2 or greater, since the rightmost number cannot be 10 or 11. By clue 4, the middle number in the third row minus the leftmost number in the second row equals 4. The leftmost number in the second row therefore must be 5 or less, since the middle value in the third row can be no more than 9. So, the leftmost number in the second row is 2, 3, 4, or 5. If the leftmost number in the second row were 2, the other number in the second row would be 9 (clue 1), the middle number in the third row would be 6 (clue 4), and the top number would be 5 (5).

By clue 2, the sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10. Since the numbers left are 0, 1, 3, 4, 7, and 8, none of 4, 7, and 8 could be in the third row with 6 because no combination of remaining numbers would add to a sum 10 greater than the second row sum. If the 3 were in the second row with 6, then 4, 7, and 8 would sum to 19--but the odd 1 couldn't go in either row, so 3 couldn't be in the second row either. Adding the 0 and 1 to 6 would give 7, but the fourth row would add to 22, contradicting clue 2. Therefore, the 2 cannot work as the leftmost value of the second row. If 4 were the leftmost number in the second row, 7 would be the rightmost (1), 8 would be the number in the middle of the third row (4), and 3 would be the number on top the Number Pyramid (5). Again by clue 2, the sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10, using 0, 1, 2, 5, 6, and 9. If any of 5, 6, or 9 were in the second row with 8, there would be no way for clue 2 to work; so those numbers would be in the bottom row, giving at least 20. Then the 2 would have to go in the second row with 8 to get 10--but the odd 1 cannot fit. So, the leftmost number in the second row isn't 4. If the leftmost number in the second row were 5, 6 would be the rightmost (1), 9 would be the number in the middle of the third row (4), and 2 would be the number on top the Number Pyramid (5), leaving 0, 1, 3, 4, 7, and 8. Again by clue 2, the sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10. None of 3, 4, 7, or 8 could be with 9 in the third row because no combination of remaining numbers in the fourth row could satisfy clue 2. With the 0 and 1 being in the second row, however, the sum of 10 would be 12 short of the 3, 4, 7, and 8 in the bottom row, contradicting clue 2. So, the leftmost number in the second row isn't 5 and must be 3. Then 8 would be the rightmost (1), 7 would be the number in the middle of the third row (4), and 4 would be the number on top of the Number Pyramid. Again by clue 2, the sum of the numbers in the bottom row minus the sum of the numbers in the third row equals 10. Of the remaining numbers, neither 5, 6, or 9 could be in the third row or clue 2 couldn't work, so those three numbers add to 20 in the bottom row. In order for clue 2 to be satisfied, the 1 and 2 must go in the third row to sum to 10 and the 0 must go in the bottom row to keep that sum at 20.

By clue 3, the righthand four numbers sum to 18, with 4 in the top and 8 in the second row leaving 6 for the bottom two rows. The only combination that works is 1 as the rightmost number in the third row and 5 as the rightmost number in the fourth row. By elimination, then, 2 is the leftmost figure in the third row. By clue 6, the 0 must be the number next to the 5; while the 9 must be the leftmost number and the 6 next to the 9.

Therefore, the Number Pyramid should look like this:

   4
  3 8
 2 7 1 
9 6 0 5

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: i did it!Rebecca Jaffe2005-08-30 05:14:22
i did it!Kitten2004-12-10 05:56:27
original solutionBob Williamson2004-10-23 00:34:11
SolutionsolutionDetti2004-10-05 07:52:23
SolutionsolutionDetti2004-10-05 07:51:55
re: # pyramidNate Morris-Ricci2004-09-29 14:34:23
# pyramiddavid2004-09-27 14:19:33
SolutionNo SubjectJACKIE TAYLOR2004-09-11 05:07:41
re: possible solutionRafal2004-09-10 09:09:50
possible solutionJosh2004-09-08 17:37:15
SolutionNecessary clues - part 2Rafal2004-09-05 11:13:53
Some ThoughtsNecessary cluesRafal2004-09-04 14:13:21
SolutionSolution with descriptionRafal2004-09-04 13:14:29
SolutionSolution?RandyOrton2004-08-26 11:28:08
Maybe...sarah2004-08-23 00:02:45
re: Question?dhruv2004-08-22 10:14:02
Question?Bruce Brantley2004-08-22 00:00:29
answer?terri2004-08-21 21:17:35
SolutionFull solution!dhruv2004-08-21 05:14:37
SolutionNo SubjectBJ2004-08-20 18:21:48
SolutionSolution, not computernikki2004-08-20 11:24:28
Some ThoughtsAlmost a solutionVishal Baghel2004-08-20 10:47:14
Explanation neededGamer2004-08-20 10:44:33
Solutioncomputer solutionCharlie2004-08-20 09:50:22
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (5)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (0)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2020 by Animus Pactum Consulting. All rights reserved. Privacy Information