Grades 9-10 Video Solutions 2023-2023_9-10

Video Transcription

7 different single digit numbers are written in the circles of the diagram shown with one number in each circle.  The product of the 3 numbers in each of the 3 lines of 3 numbers is the same. Which number is written in the circle containing the question mark?  So let's consider the single digit numbers that we are putting into the circles. And let's try to rule out some of them. Think about the numbers 5 and 7.  If we use the numbers 5 and 7 in the diagram, then that would mean that the product of all  3 lines must be a multiple of 5 or 7, because the product of all 3 lines is the exact same. However, we know that this isn't possible, because 5 and 7 can both only be an at most  2 of the 3 lines, and because there are no other single digit multiples of 5 or 7, then it wouldn't be possible that they have a product of 5 or 7.  Therefore, we can rule out the numbers 5 and 7. So let's consider the 2 horizontal rows that were on the top.  Because the 2 horizontal rows had the same product, then the product of the 2 horizontal rows must be a square number.  Considering the 7 digits that we are allowed to use after ruling out 5 and 7, we find that 1 times 2 times 3 times 4 times 6 times 8 times 9 is 2 to the 7 times 3 to the 4.  That means that the digit covered by a question mark in the diagram previously must either be 2 or 8 in order to ensure that the product of the 2 horizontal rows is a square number.  Suppose that the covered square were 8, then the product of each row would have to be the square root of 1, 2, 3, 4, 6, 9, which we know is 36, because we have the number 8 removed.  However, if 36 were the product of each row, then the vertical column would fail, because if 36 were the product of each row, that would imply that the vertical column's product is  also 36, but we know that no multiple of 8 is 36. Then we've ruled out all the possible numbers, and then the number with the question mark must be 2 on the bottom.  A possible solution with all numbers would be reading from right to left, in the first row 1, 9, 8, in the second row 3, 4, 6, and down the column 9, 4, 2.  In this scenario, the product of each of the columns and row would be 72.

Video Summary

The video explains a puzzle where seven different single-digit numbers are placed in circles, with each line of three circles having the same product. Numbers 5 and 7 are ruled out because no three-line products could universally include them. The puzzle requires the two horizontal rows to produce a square product, and considering allowed numbers after ruling out 5 and 7, the missing number under the question mark is determined to be 2. This allows a setup where each line's product is 72, with the correct configuration being: top row (1, 9, 8), middle row (3, 4, 6), and a column (9, 4, 2).

Keywords

puzzle

single-digit

product

numbers

configuration