Arithmetic Drill

How it works

The drill deals a fixed deck of 20 cards, one at a time. Every card is a freshly generated problem from one of twelve categories: long addition (three-digit plus three-digit), long subtraction, two-digit by two-digit multiplication, fast division (always exact), decimal addition, subtraction, and multiplication, and five fraction operations - add, subtract, multiply, divide, and reduce to lowest terms. Type your answer and press enter; there is no second attempt on a card.

You can play solo or race a ghost - a recording of a real past run on the exact same deck. The deck is rebuilt from a shared seed, so the ghost sees the same cards in the same order and claims each card at the moment its player actually answered it. If the ghost buzzed in wrong on a card, it eats the penalty and the card stays live for you.

Each card has a 10 second timeout so one stumper cannot stall the race. A card nobody claims simply expires. Fraction answers accept either "a/b" or a decimal, but reduce-fraction cards require the exact reduced form - 4/8 does not pass for 1/2. Decimal answers are graded with a small numeric tolerance, whole-number answers by exact match.

How scoring works

You score +1 for taking a card with a correct answer and -1 for buzzing in wrong. A wrong answer forfeits the card immediately, the same as being beaten to it. A timed-out card scores nothing for anyone.

In ghost mode the ghost scores +1 for each card it claims first with a correct answer and -1 where its recorded run answered wrong. The race ends when the 20th card resolves; there is no wall clock, so a fast run simply finishes fast.

Your run is saved with its seed and per-card times, which is what lets a future player race you as a ghost on the identical deck.

Work left to right, not right to left

School arithmetic teaches right-to-left with carries, which is the slowest possible order under a clock because you cannot say any digit until you have finished all of them. For a card like 738 + 486, go left to right: 700 + 400 = 1100, then 38 + 86 = 124, giving 1224. You hold one running total instead of three carry flags, and the leading digits - the ones that catch a typo - come first.

For subtraction, use complements instead of borrowing. 903 - 458 is easier as 903 - 458 = 903 - 500 + 42 = 445. Round the subtrahend to a clean number, subtract, then add the difference back. Borrowing across a zero is exactly where slow, error-prone runs die; complements never borrow.

For two-digit multiplication, anchor on a round number: 47 x 62 = 47 x 60 + 47 x 2 = 2820 + 94 = 2914. When one factor is near a landmark, use the difference instead: 49 x 38 = 50 x 38 - 38 = 1900 - 38 = 1862. One adjustment step beats a full long-multiplication grid every time.

Make the fraction cards automatic

The denominators here are always drawn from a small set (2, 3, 4, 5, 6, 8, 9, 10, 12), so the useful common denominators are a short list you can memorize: pairs from this set almost always meet at 12, 24, or the product. For addition and subtraction, cross-multiply mechanically - a/b + c/d = (ad + bc)/bd - then reduce. Do not hunt for the least common denominator under a clock; the cross-multiplied form plus one reduction is faster than being clever.

Multiplication and division should be cancel-first. For 3/8 x 4/9, cancel the 4 into the 8 and the 3 into the 9 before multiplying: 1/2 x 1/3 = 1/6. Division is one flip away: a/b divided by c/d is a/b x d/c, then cancel. If you multiply first and reduce later you are doing gcd work on bigger numbers for no reason.

Reduce-fraction cards are pure gcd recognition, and the generator builds them by scaling a reduced fraction by 2 through 6. So check the small divisors in order - both even? Halve. Digit sums divisible by 3? Divide by 3. Then 5. Two passes of that reaches lowest terms on every card this game can deal. Remember the grader here demands the exact reduced fraction, so finish the reduction.

Race the ghost, not the deck

In ghost mode the winning move changes: you are not maximizing correctness, you are claiming cards before a specific opponent whose speed on each card is fixed. A ghost that averaged 6 seconds a card can be beaten by someone slower overall who is fast on the easy categories - the addition and division cards resolve in 2 to 3 seconds for a practiced player, and every one you take is a point the ghost cannot have.

The -1 for a wrong buzz means guessing is genuinely negative expected value. A wrong answer both costs you a point and hands the card's point opportunity away, a two-point swing against sitting on your hands. If a card is not coming to you inside a few seconds, it is usually better to let a strong ghost take it and be ready for the next flip than to fire a guess.

Solo runs are where you build the deck sense: after each run, note which of the twelve categories cost you the most time. Most players lose the race on exactly two categories - usually two-digit multiplication and fraction addition - and targeted practice on those moves the whole run more than grinding the categories you already own.

A worked example

Card: 67 x 84. Left-anchor it: 67 x 84 = 67 x 80 + 67 x 4. First piece: 67 x 8 = 536, so 67 x 80 = 5360. Second piece: 67 x 4 = 268. Sum left to right: 5360 + 268 = 5628.

Sanity check in one beat before pressing enter: 67 is near 70 and 84 is near 80, so the answer should be near 5600. It is. Type 5628, enter, card claimed - roughly four seconds, against a 10 second timeout.

Common mistakes

• Buzzing in with a guess when unsure - a wrong answer scores -1 and forfeits the card, so a guess is a two-point swing worse than passing.

• Doing multiplication right to left with carries instead of anchoring on a round number and adjusting.

• Submitting an unreduced fraction on a reduce-fraction card - the grader requires the exact lowest-terms form, so 4/8 fails where 1/2 passes.

• Hunting for least common denominators on fraction addition instead of cross-multiplying and reducing once.

• Forgetting the 10 second per-card timeout and settling into a long calculation the card will not wait for.

Why interviews test this

Timed arithmetic under pressure is the front door of trading interviews. Zetamac-style speed tests and firm-specific equivalents at market-making firms filter on exactly this skill: many small computations, fast, with a penalty for errors. The point is not that traders do long division all day - it is that fluent arithmetic frees working memory for the actual decision.

The race format here mirrors the culture: in a trading interview, an answer that is fast and right beats one that is slow and right, and one that is fast and wrong is worse than silence. That is precisely the +1 / -1 / pass structure of this drill.