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Knowledge Hub · Article 347 · Maths

Hard math questions: 20 genuinely hard problems with full solutions, from Grade 6 to SAT level, across number, algebra, geometry, counting and probability, with the trick each one turns on, what makes a question hard, and a printable set to try before reading the answers

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Article 347 · Maths

"Hard math questions" is searched about 2,500 times a month in the United States and 800 in Canada, by students who want to be stretched and by parents looking for something beyond the worksheet, and "hardest SAT math questions" by students aiming for the top of the Math section. This page gives twenty questions that are hard in the useful sense, where the difficulty is finding the idea rather than grinding the arithmetic, arranged roughly from Grade 6 to SAT and competition level, each with a full solution and the idea named. The printable has the questions alone on page 1, so they can be tried properly first, and the solutions on page 2.

Download the 20 hard questions with solutions (PDF) (A4, questions on page 1, full solutions with the idea for each on page 2, free).

What makes a question hard

A hard question hides the idea. The arithmetic in all twenty below is Grade 6 level; what each one tests is whether you can see that a perimeter-and-area question is about factor pairs, that a "last digit of a huge power" question is about cycles, that a remainder puzzle is about the LCM. The twenty ideas here (pairing, cycles, LCM, factor pairs, complementary counting, symmetry, rates as fractions of a job, working backwards) cover most of what competition maths and the hardest SAT items ask at this level. Learn the ideas, not the problems.

The twenty questions

Try each before scrolling to its solution. Suggested time: 3 minutes each for 1 to 10, 5 minutes for 11 to 20.

  1. What is 1 + 2 + 3 + ... + 100?
  2. What is the smallest positive whole number that leaves remainder 2 when divided by 3, remainder 3 when divided by 4, and remainder 4 when divided by 5?
  3. What is the last digit of 7²⁰²⁶?
  4. A rectangle has whole-number sides, a perimeter of 30 and an area of 56. What are its sides?
  5. How many diagonals does a decagon (10 sides) have?
  6. Two fair dice are rolled. What is the probability that the total is 7?
  7. A train 120 m long passes a signal post in 6 seconds. How long does it take to pass completely through a platform 180 m long?
  8. If x + 1/x = 3, what is x² + 1/x²? And x³ + 1/x³?
  9. What is the angle between the hands of a clock at 3:30?
  10. Twelve people at a meeting each shake hands once with every other person. How many handshakes?
  11. A father is four times as old as his son. In 20 years he will be twice as old. How old are they now?
  12. How many three-digit numbers have digits that add up to 5?
  13. A right-angled triangle has legs 5 and 12. What is the radius of the circle inscribed in it?
  14. A can paint a room in 6 hours and B in 9 hours. How long do they take together?
  15. The average of five numbers is 20. One number is removed and the average of the remaining four is 18. What number was removed?
  16. 20 litres of a 30% acid solution are to be diluted to 20% by adding water. How much water?
  17. The sequence 2, 6, 12, 20, 30, ... What is the 10th term, and what is the rule?
  18. What is the sum of the interior angles of a 12-sided polygon, and what is each interior angle of a regular pentagon?
  19. How many whole numbers from 1 to 1,000 are divisible by 3 or by 5?
  20. (SAT) If 3x − 2y = 7 and x + y = 4, what is the value of x − y?

The solutions, with the idea each turns on

1. 5,050. Idea: pairing. 1 + 100, 2 + 99, ..., 50 + 51: fifty pairs of 101. 50 × 101 = 5,050. In general 1 + ... + n = n(n + 1) ÷ 2.

2. 59. Idea: look at what is missing. Each remainder is one less than the divisor (2 of 3, 3 of 4, 4 of 5), so the number is one less than a common multiple of 3, 4 and 5. LCM = 60, so N = 60 − 1 = 59. Check: 59 ÷ 3 = 19 r 2, 59 ÷ 4 = 14 r 3, 59 ÷ 5 = 11 r 4.

3. 9. Idea: cycles. Last digits of 7ⁿ: 7, 9, 3, 1, repeating every 4. 2026 ÷ 4 = 506 r 2, so the last digit is the second in the cycle: 9.

4. 7 and 8. Idea: factor pairs. Perimeter 30 means the sides add to 15; area 56 means they multiply to 56. Factor pairs of 56: 1 × 56, 2 × 28, 4 × 14, 7 × 8. Only 7 + 8 = 15. (Or solve s² − 15s + 56 = 0.)

5. 35. Idea: count from each vertex, then halve. From each of 10 vertices, diagonals go to 7 others (not itself or its two neighbours): 10 × 7 = 70, but each diagonal is counted from both ends, so 35. Formula: n(n − 3) ÷ 2.

6. 1/6. Idea: list the outcomes. 36 equally likely pairs. Totals of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1): six of them. 6/36 = 1/6, the most likely total.

7. 15 seconds. Idea: the train's own length counts. Passing a post, the train travels its own length: 120 m in 6 s, so 20 m/s. To clear a 180 m platform the front must travel 180 + 120 = 300 m: 300 ÷ 20 = 15 s.

8. 7, then 18. Idea: square the given equation. (x + 1/x)² = x² + 2 + 1/x² = 9, so x² + 1/x² = 7. Then (x + 1/x)(x² + 1/x²) = x³ + 1/x³ + x + 1/x, so 3 × 7 = x³ + 1/x³ + 3, giving x³ + 1/x³ = 18. Never solve for x.

9. 75°. Idea: the hour hand moves too. The minute hand at 30 minutes points at 6: 180°. The hour hand at 3:30 is halfway from 3 to 4: 3.5 × 30° = 105°. Difference 180 − 105 = 75°. (The hour hand moves 0.5° a minute.)

10. 66. Idea: pairs, counted once. Each of 12 people shakes 11 hands: 12 × 11 = 132, but each handshake involves two people, so 66. Formula: n(n − 1) ÷ 2, the same as "choose 2 from 12".

11. Son 10, father 40. Idea: set up both times. Let the son be s; the father is 4s. In 20 years: 4s + 20 = 2(s + 20) → 4s + 20 = 2s + 40 → 2s = 20 → s = 10. Father 40. Check: in 20 years, 60 and 30.

12. 15. Idea: organised counting. The hundreds digit a is 1 to 5 (the digits sum to 5), and the other two digits b + c = 5 − a with b, c from 0 to 9. a = 1: b + c = 4, five ways (0+4 ... 4+0). a = 2: four ways. a = 3: three. a = 4: two. a = 5: one (500). Total 5 + 4 + 3 + 2 + 1 = 15.

13. 2. Idea: the inradius formula for a right triangle. Hypotenuse √(25 + 144) = 13. For a right-angled triangle, r = (a + b − c) ÷ 2 = (5 + 12 − 13) ÷ 2 = 2. (Or area = r × semi-perimeter: 30 = r × 15.)

14. 3.6 hours (3 h 36 min). Idea: rates as fractions of the job. A does 1/6 of the room per hour, B does 1/9. Together 1/6 + 1/9 = 3/18 + 2/18 = 5/18 per hour. Time = 18/5 = 3.6 hours. Never average the times.

15. 28. Idea: work with totals, not averages. Five numbers averaging 20 total 100. Four numbers averaging 18 total 72. The removed number is 100 − 72 = 28.

16. 10 litres. Idea: the acid does not change. 30% of 20 L is 6 L of acid. After adding w litres of water, 6 ÷ (20 + w) = 0.2, so 20 + w = 30, w = 10.

17. 110; the rule is n(n + 1). Idea: look at differences, then factor. Differences 4, 6, 8, 10 (growing by 2: quadratic). Each term is n × (n + 1): 1×2, 2×3, 3×4, 4×5, 5×6. The 10th is 10 × 11 = 110. (Also twice the triangular numbers.)

18. 1,800° and 108°. Idea: split into triangles. A polygon with n sides splits into n − 2 triangles from one vertex, so the angle sum is (n − 2) × 180°. 12 sides: 10 × 180 = 1,800°. A pentagon: 3 × 180 = 540°, and a regular one has 540 ÷ 5 = 108° at each corner.

19. 467. Idea: inclusion-exclusion. Multiples of 3 up to 1,000: 333. Multiples of 5: 200. Multiples of both, that is of 15: 66. Counted twice, so subtract once: 333 + 200 − 66 = 467.

20. 2. Idea: solve the system, or spot the combination. From x + y = 4, y = 4 − x. Substitute: 3x − 2(4 − x) = 7 → 5x − 8 = 7 → x = 3, y = 1. x − y = 2. (The SAT often wants a combination like x − y directly; here it is only a step away.)

The ideas, collected

Idea Questions Where it shows up again
Pair things up 1, 10 arithmetic series, combinations
Cycles and remainders 2, 3 modular arithmetic, calendar problems, LCM puzzles
Factor pairs 4 quadratics by factoring, Diophantine problems
Count from each vertex and halve 5, 10 handshakes, diagonals, chords
List the sample space 6 every probability question with dice, coins, cards
Include the object's own length 7 trains, tunnels, bridges
Square the equation 8 symmetric expressions, SAT "what is x² + y²"
The hour hand moves 9 all clock-angle problems
Set up both time points 11 every age problem
Organised counting 12 digit problems, paths on a grid
Area = r × semi-perimeter 13 incircles, triangle geometry
Rates as fractions of the job 14 pipes, workers, printers
Totals, not averages 15 every "average changes when" problem
The unchanging quantity 16 mixtures, concentration, alloys
Differences, then factor 17 sequences, nth-term rules
Split into triangles 18 polygon angles
Inclusion-exclusion 19 "divisible by a or b", Venn problems
Combination of equations 20 SAT systems, "find x + y"

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics and teaches Grade 6 to 12, KS3 to A-level, SAT Math and competition preparation (AMC 8 and 10, UKMT Junior and Intermediate, Australian Mathematics Competition) every week: hard problems taught by the idea, with the child attempting first and the solution studied second, and a notebook of ideas built over the term. The SAT page explains the exam preparation; the free 30-minute first class is a real lesson on a problem the child has been stuck on. Related pages: prime numbers to 100 and the sieve, square numbers and the patterns and the SAT score calculator.

Questions parents ask

1What makes a math question hard?
Usually not the arithmetic. A hard question hides which idea to use: it gives a perimeter and an area and expects you to think of factor pairs, or asks for a remainder and expects you to notice a pattern in the last digits. Once the idea is found the working is short. Questions that are merely long (many steps, big numbers) feel hard but are different: they test care, not insight.
2What is the hardest type of math question on the SAT?
By wrong-answer rates, the hardest SAT Math questions are usually the last few in each module: systems of equations with a parameter (for what value of k does the system have no solution), quadratics with a condition on the roots (the discriminant), circle equations needing completing the square, and word problems with percentages applied twice. None uses maths beyond Algebra 2; they combine two ideas in one question.
3How do you get better at hard math questions?
Work the problem for a fixed time before reading the solution, then study the solution for the idea rather than the steps, and write the idea down in a line ("perimeter and area: think factor pairs"). Redo the problem a week later from the idea alone. Twenty problems studied this way teach more than two hundred done once, because hard questions repeat their ideas far more than their surface stories.
4What is the sum of the numbers from 1 to 100?
5,050. Pair the first and last numbers: 1 + 100 = 101, 2 + 99 = 101, and so on, fifty pairs of 101, so 50 × 101 = 5,050. The general formula for 1 + 2 + ... + n is n(n + 1) ÷ 2, and the story is that Gauss found it as a schoolboy.
5What is the last digit of 7 to the power 2026?
9. The last digits of the powers of 7 cycle with period 4: 7, 9, 3, 1, 7, 9, 3, 1 ... Divide 2026 by 4: remainder 2, so the last digit is the second in the cycle, 9. Every "last digit of a huge power" question is a cycle question.

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