Logical Squares:
Find the missing value!
problem 1:
3 24 4
5 120 100
1 0 ?
problem 2:
? 2 5 8
7 9 11 6
13 4 17 18
19 21 23 24
SPOILER
Solution for problem 1:
-----------------------
This problem is much too small to make a serious guess.
Even the problem composer gives two alternative values.
Solution: Relation between the first and the middle columns
model 1:
first, middle
t t^3-t
0 0
1 0
2 6
3 24
4 60
5 120
6 210
7 336
model 2:
first, middle
t ((t^2+4t-1)/4)^2 - 1
1 0
3 24
5 120
7 360
Solution: Relation between the middle and the last columns
model 1:
middle, last
t (t/12)^2
Therefore we get for t=0 the missing value ? = 0.
model 2:
middle, last
t t-20
Therefore we get for t=0 the missing value ? = -20.
Solution: Relation between the first and the last columns
model 1:
first, last
t 48t - 140
1 -92
3 4
5 100
Therefore we get for t=1 the missing value ? = -92.
model 2:
first, last
t (4t-10)^2
1 36
3 4
5 100
Therefore we get for t=1 the missing value ? = 36.
Solution: linear dependence
Determine '?' such that the three vectors become
linear dependent.
| 3 24 4 |
| 5 120 100 | = 0 Therefore '?' = -8.
| 1 0 -8 |
This is the relation 48x - 7y + 6z = 0.
Reference:
R. Zehl;
Denken mit Spass 2, Die neuesten trend-Nuesse,
ORAC Verlag, Wien, 1984, ISBN 3-85368-983-3, Problem 11
Solution for problem 2:
-----------------------
Read the numbers from left to right:
? 2 5 8 7 9 11 6 13 4 17 18 19 21 23 24
Drop every second number:
? 5 7 11 13 17 19 23
Solution 1:
We have the sequence of odd primes: '?' = 3
Solution 2:
We have the numbers 6n+-1 in ascending order: '?' = 1
Reference:
New Beetle Riddle, Autohaus Menkel, Frankenberg (Eder), Germany