Problem 2 · 1998 AJHSME
Easy
Algebra & Patterns
custom-operation
If acbd = a·d − b·c, what is the value of 3142 ?
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Answer: E — 2.
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Hint 1 of 2
A made-up symbol can't trick you — it's just a recipe. Read off which numbers play the roles of a, b, c, d, then follow the recipe exactly.
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Hint 2 of 2
The rule a·d − b·c is a criss-cross: multiply the two corners on one diagonal, subtract the product of the other diagonal.
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Approach: follow the recipe (criss-cross of the corners)
- Match the positions to the rule: a = 3 (top-left), b = 4 (top-right), c = 1 (bottom-left), d = 2 (bottom-right). The rule wants a·d − b·c.
- So 3·2 − 4·1 = 6 − 4 = 2.
- Why this transfers: contests love inventing a brand-new symbol just to see if you'll calmly substitute into the definition. There's nothing to memorize — locate the inputs, run the recipe. (This particular criss-cross is the 2×2 determinant you'll meet again later.)
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