Expert: Josh - 9/24/2006

Question
Hi Josh, hope you can help me.  I am stuck with this word problem:
Eric and his friends practice multiplying by using dominoes.  Each half of a domino has dots on it to show a number from 0 to 6.  The students use the two numbers on a domino as factors.
a.What is the greatest product you can make from numbers on dominoes?
c.Eric reasons that he has to know the answers for 0x0,0x1,0x2,0x3,0x4,0x5,0x6,1x0,1x1 and so on.  Because there are seven different numbers, 0,1,2,3,4,5 and 6, that can occur on each hlaf of the domino, he reasons that he needs to know 49 different answers.  This is too many.  What did he forget?

Answer
Hi Krystal,

a) greatest product when two such numbers are multiplied together is 6*6
c) 7*7=49 unique combinations would result if the seven numbers on the left and right are different. In this case, the left and right halves both take numbers from {0,1,2,3,4,5,6}. So, the order is interchangeable. e.g., 0*4 is the same as 4*0. Draw a table to see this. Hint: Let the rows and columns be indexed by 0,1,2,...6.

   0    1    2    3    4    5    6
0 (0,0)(0,1)(0,2)(0,3)(0,4)(0,5)(0,6)
1 (1,0)(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)
2 (2,0)(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)
3 (3,0)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)
4 (4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)
5 (5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)
6 (6,0)(6,1)(5,2)(6,3)(6,4)(6,5)(6,6)

It is clear that we need the entries in the upper right triangle. Thus, there are 7+6+5+4+3+2+1 entries that he needs to know. This answer may be found using the arithmetic sum formula, (N/2)*(A+L), where N=7 is the number of terms, A=1 is the value of the first term, L=7 is the value of the last term. Eric needs to know (7/2)*(1+7)=28 terms.