Grades 11-12 Video Solutions 2022-2022_11-12

Video Transcription

The numbers 1 to 10 are placed, once each, in the circles of the figure shown. The sum of the numbers in the left column is 24,  and the sum of the numbers in the right column is also 24, and the sum of the numbers in the bottom row is 25. What number is in the circle containing the question mark?  So here's the figure, drawn a little bit bigger. So, we know that we use the numbers from 1 to 10 exactly once, so the sum of every single circle has to be equal to 55.  We also know that the left and the right are 24, and so if we subtract the left and the right from the total, we can calculate the sum of the two middle ones, which has to be 7.  So that means, in these two middle spots, you have any number from 1 to 6, and in the other spot you have to have 7 minus whichever one's in the question mark.  So now let's look at the bottom row. If I max out the two corners by putting a 9 and a 10 in there, we need to somehow get a 25, and we can use something at maximum 6.  Well, it turns out 6 does the trick, and anything less than 6 isn't going to reach 25. So that means this middle circle has to be 6, and the top circle has to be 1,  which isn't any of these options, so it has to be none of the previous.

Video Summary

In the given puzzle, numbers 1 to 10 are used to fill circles with specific column and row sums. The total sum of the numbers in the circles is 55. The sums for the left and right columns are both 24, leaving the numbers in the middle column summing to 7. After trying different combinations, it is determined that the middle circle in the bottom row must be 6 to achieve the sum of 25 with maximum corner values. Therefore, the circle with the question mark must contain the number 1 to satisfy all conditions, but this wasn't an option, leading to "none of the previous" as the answer.

Keywords

puzzle

numbers

sums

combinations

solution