the sum of the four digits on each side of the triangle is the same for all three sides? (Each digit is used exactly once.) Can you figure out how to arrange the digits so that the side sums are equal and at a minimum or maximum? There is more than one possible solution.

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Here's what I came up with:

The minimum total for each side is 13. One arrangement of the numbers, reading left to right, top to bottom, is 0, 8, 6, 4, 9, 5, 1, 7, 3, and 2. On the left side, that results in the sum 0+8+4+1=13. On the right side, 0+6+5+2=13. The sum across the base is 1+7+3+2=13. The 9 takes the center circle, and so it is not used in any of the sums.

One arrangement that gives the maximum total of 23, again reading left to right, top to bottom, is 9, 1, 3, 5, 0, 4, 8, 2, 6, and 7. Therefore, on the left, the numbers 9, 1, 5, and 8 add up to 23. On the right, the numbers 9, 3, 4, and 7 add up to 23. And across the base, the numbers 8, 2, 6, and 7 add up to 23.

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You solved both problems, nicely done.

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I came up with:

9 at the top center

1 and 4: next row/ 4 on right-hand side

503 next

8267 across the bottom row

My approach was to get the biggest number out of the way by putting the 9 on the top, then the 8 at the next "end" circle, and then the 7. From there, and using the "middle" (not the smallest) numbers, someone can figure out fairly easily what the maximum sum is ever going to be for the rows. Keeping the 0 for last (in case I needed it) (and with the larger numbers already in place, and keeping in mind that the larger the number; the smaller the "additional" numbers are going to be... It was fairly easy to make them add up to the 23 I'd gotten after figuring out what that maximum "add-up" was going to be.

The biggest numbers make an easy guide for the rest of it. By the time you get to the smallest numbers (and 0) there's less flexibility but more info to go by already in place.

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Thanks for the great explanation of your thought process to solve this problem. A lot of people get stuck when they can't figure out what the sum should be after placing the center and corner numbers.