- What is a diamond problem?
- A diamond problem is a square drawn on its point and divided into four triangles. The two side cells hold a pair of numbers; the top cell holds what they multiply to, and the bottom cell holds what they add to. You're given two of the four and asked for the rest. It shows up in the CPM Core Connections curriculum around Algebra 1, usually a chapter or two before factoring, and the reason it exists is that 'find two numbers with this product and this sum' is precisely the step you need to factor x² + bx + c later. The diamond is the drill; factoring is the thing it's drilling.
- How do you solve a diamond problem when you're given the product and the sum?
- By hand, you list the factor pairs of the product and check which pair adds to the sum. For a product of 24 and a sum of 10: 1 and 24 add to 25, 2 and 12 add to 14, 3 and 8 add to 11, 4 and 6 add to 10 — there it is. When the numbers don't cooperate, the reliable route is algebra: the two numbers are the roots of x² − Sx + P = 0, so you take the discriminant S² − 4P, square-root it, and split the sum either side. That's the method this page shows in the working, because it never depends on spotting a lucky pair.
- What if the product is negative?
- Then one number is positive and the other is negative — that's the single most useful sign rule in the whole topic. Whether the sum is positive or negative tells you which of the two is bigger in size: a product of −24 with a sum of −2 gives 4 and −6, because the negative one has to be the larger of the pair to drag the total below zero. If the product is positive and the sum is negative, both numbers are negative. Students lose more marks to sign slips on these than to arithmetic, so the working on this page always shows the signs explicitly rather than tidying them away.
- What if only one number and the sum are given?
- Subtract. If the left cell is 5 and the sum is 2, the right cell is 2 − 5 = −3, and the product is 5 × −3 = −15. Given one number and the product instead, divide: 56 ÷ 7 = 8, then add for the sum. Both of these are far quicker than the product-and-sum direction, which is why it's odd that most solvers online don't accept them — a worksheet hands you these constantly. Type into whichever two cells you actually have and this page picks the right route on its own.
- Why do I sometimes get a decimal instead of two whole numbers?
- Because no whole-number pair exists for those two values. A product of 1 and a sum of 3 gives 2.6180 and 0.3820 — the square of the golden ratio and its reciprocal, as it happens — and the page flags that the pair isn't whole. Diamond problems are almost always designed to come out as integers, so a decimal answer is usually a signal that a digit or a sign got typed wrong somewhere. Read it as a prompt to check your inputs rather than as the answer you were meant to find.
- Why does it sometimes say no two real numbers do that?
- Because you've asked for a product that's too big for the sum you gave. Two numbers adding to S can only multiply to at most S²/4, which happens when they're equal halves — so a sum of 2 caps the product at 1, and asking for 10 is impossible over the real numbers. Algebraically the discriminant S² − 4P has gone negative, and the roots are a complex pair. The page tells you the largest product your sum can reach, because that number usually points straight at the typo.
- Does it handle fractions?
- Yes, and exactly. Type 1/2 and 1/3 and you get a product of 1/6 and a sum of 5/6 — not 0.16666666666666666, which is what you'd see from a page doing the arithmetic in ordinary decimals. Fractions go in as fractions and come back as fractions; type decimals and you get decimals back, so 0.1 and 0.2 give 0.02 and 0.3 rather than the 0.020000000000000004 that floating-point produces. CPM diamond problems use fractions often enough that this is the difference between a usable page and a confusing one.
- How is this connected to factoring a quadratic?
- Directly. To factor x² + 7x + 12 you need two numbers that multiply to 12 and add to 7 — a diamond problem with the product on top and the coefficient of x on the bottom. Solve it (3 and 4) and the factorization falls out: (x + 3)(x + 4). For a quadratic with a leading coefficient, like 2x² + 7x + 3, the diamond takes the product a·c on top and b on the bottom, which is the 'split the middle term' method. If you want the full factorization rather than the pair, the quadratic formula calculator takes it from here.
- Can I fill in three cells?
- You can type in all four if you like — the page keeps the two you touched most recently and re-solves around those, recomputing the rest. That's deliberate: when you're correcting a number you've already entered, you want the page to follow your latest thinking rather than lock you into your first two entries or throw a contradiction on screen. Clear a cell and the derived answers clear with it.
- Is anything I type sent anywhere?
- No. The whole thing runs in your browser, so nothing leaves the page and closing the tab clears it. There's no account, no limit on how many diamonds you solve, and the working isn't held back for a subscription — the method is the part worth having, so it's the part that's on screen.
