MAX AND MIN
Max and min problems, the extremes follow rules.
Four classic max-min results in one page: with a fixed sum the product grows as the numbers get closer; with a fixed product the sum shrinks the same way; splitting a number for the largest product means using as many 3s as possible; and the largest rectangle for a given perimeter is a square. Each comes with step-by-step reasoning and random practice.
Results will appear here
Everything runs locally in your browser
How to use
Pick a type, enter the condition, and solve. Fixed sum: the closest pair wins. Fixed product: all factor pairs are listed and the closest pair gives the least sum. Splitting: as many 3s as possible, swapping a 3+1 for 2+2 when there is a remainder of 1. Perimeter: half of half the perimeter gives the square. Every step includes a check you can verify.
常见问题
Why does a closer pair give a bigger product at a fixed sum?
For numbers a and S-a, the product a(S-a) = -(a-S/2)^2 + S^2/4, largest at a = S/2. Intuition: among rectangles with the same perimeter, the closer length and width are, the larger the area - the square is the extreme. Spreading apart toward 1 shrinks the product.
Why split mostly into 3s?
6 = 3+3 gives product 9, while 2+2+2 gives only 8, so 3 beats 2 at the same size. But 4 = 3+1 loses to 2+2 (3x1 < 4), so a remainder of 1 is merged with a 3 into 2+2. Rule: remainder 0 - all 3s; remainder 1 - two 2s plus 3s; remainder 2 - one 2 plus 3s.
Why is the square largest for a fixed perimeter?
With length + width fixed, the product is largest when they are equal (see the first FAQ). So a rectangle with perimeter C is largest at side C/4, area (C/4)^2.
Is any data uploaded?
No. All solving and question generation run locally in the browser.