How many practice problems should you do a day?

There is no good number, and anyone who gives you one without asking two questions first is guessing. Ten a day is too many if every one of them is a fifty-minute proof, and far too few if every one is a problem you could already do in your sleep.

The count is an output, not a target. What you actually control is how long you sit down for and what you do with each problem after you have tried it. Get those right and the number of practice problems per day falls out on its own. Aim at the number directly and you will hit it by choosing problems that are easy enough to hit it with.

What a problem is for

A problem you solve at once, with no hesitation, confirms something you already knew. That has some value on the day before an exam. On an ordinary Tuesday it mostly spends time.

The problems that change you are the ones you get wrong, or get right slowly, or get right by a route you would not have found if you had not been stuck first. And the change does not happen while you are attempting them. It happens afterwards, when you work out which step broke and what you should have noticed on the first read.

So a practice problem has two costs: the attempt, and the review when it goes wrong. A plan that counts only the attempts is budgeting for half of the work.

Size the session, not the count

Write it as arithmetic, because it is arithmetic. Say you have $T$ minutes, an attempt takes about $a$ minutes, you miss a fraction $m$ of the problems, and each miss takes $r$ minutes to review properly. Then the number you can honestly do is

$$N = \left\lfloor \frac{T}{a + m r} \right\rfloor.$$

Take AMC-style questions. The AMC 10 and AMC 12 give you 75 minutes for 25 questions, so three minutes an attempt is the real pace. Give each miss ten minutes of review, which is enough to redo it from a blank page and write down the step that broke, and sit for an hour:

• miss one in twenty, and $N = 17$;
• miss one in five, and $N = 12$;
• miss one in two, and $N = 7$.

Read that table the other way round and it tells you something useful. If you are getting through seventeen or more in an hour, you are missing almost nothing, which means the set is below you. If you are getting through seven, you are working at the edge of what you can do, which is where the improvement is. A day with fewer problems is often the better day. The count falls when you move up a level, and that is the plan working, not a slump.

The one rule the formula enforces: never start a problem you will not have time to review. A miss left unreviewed is an attempt you paid for and learned nothing from.

What forty-five minutes looks like

A concrete session, for a student working in the middle band of AMC papers:

1. Six timed problems, three minutes each: eighteen minutes.
2. Review the misses. At a miss rate of one in three that is two problems, at ten minutes each: twenty minutes.
3. Recognition reps with what is left. Seven minutes buys four of them at ninety seconds each.

Step 3 needs explaining, because it is the part most plans leave out.

Recognition reps are the cheap half

Most lost marks on a contest are not problems where you knew the method and fumbled it. They are problems where you never worked out which method it was. The paper does not print "pigeonhole" at the top of the question, and working that out is a separate skill from executing it.

A recognition rep trains only that skill. Read a problem, write the first line of a solution and the name of the idea it rests on, and stop. Do not solve it. At about ninety seconds each, twenty of them take thirty minutes, which is roughly what two full attempts with review cost. The pigeonhole recognition guide runs the same drill for one technique: pigeons, boxes, why a collision finishes it, and nothing else.

Here is a four-problem set done that way.

• Among any 13 people, two share a birth month. First line: twelve months are the boxes. Idea: pigeonhole.
• A number is divisible by 9 exactly when its digit sum is. First line: $10 \equiv 1 \pmod 9$, so every power of ten is too. Idea: modular arithmetic.
• For every real $x$, $x^2 + 1 \ge 2x$. First line: the difference is $(x-1)^2$. Idea: a square is nonnegative, the seed of AM-GM.
• A set with $n$ elements has $2^n$ subsets. First line: each element is in or out, independently. Idea: count by a sequence of choices.

Four problems, four different ideas, about six minutes. None of them is solved on the page, and all of them are recognised, which is the part the contest actually tests first.

Mix the set, or the count is lying to you

Now suppose you did ten problems from a single technique page instead. Every one of them is a pigeonhole problem, and you knew that before you read the first word. You practised executing pigeonhole ten times and recognising it zero times.

That is why the same ten problems are worth more in a mixed order than in a block. A mixed set makes you decide, on every problem, what kind of problem it is. A blocked set makes that decision for you, once, at the top of the page. The blocked session feels better, because you get more of them right, and that feeling is exactly what the count rewards and what the contest does not.

Blocks have one proper use: the first time you meet a technique, a short run of it teaches the mechanics. After that, mix.

Olympiad problems change the arithmetic, not the logic

For proof problems the attempt alone can take most of an hour, and the review of a failed proof takes longer than the review of a wrong number. Put those into the same formula and one or two full problems a day is the honest answer for most students at that level. That is not too few. It is what the work weighs.

The recognition reps matter even more here, because at olympiad level a wrong first idea can cost the whole paper. One full proof and ten first lines is a better day than three proofs, most of the time, for the same reason as above: the first lines are where you train the part you cannot fix once the clock is running.

The short answer

• As many as you can attempt and review in the time you have. Review is not optional, so budget it first.
• Fewer than you think, if you are working at the right level. A number that keeps rising means the problems stopped being hard.
• Always mixed, after the first session on a technique.
• Topped up with first-line recognition reps, which are the cheapest practice there is.

To find your band instead of guessing it, start with the diagnostic. The problem archive states an expected time on every problem, which gives you $a$ in the formula rather than an estimate of it. The daily problem posts one a day in three tiers with no account needed, which is a floor for the days you have no session at all. And if you are planning a whole month rather than a single day, the one-month AMC 12 plan is built on the same idea: cut what cannot pay, and review everything you get wrong.