Getting your questions ready
Getting your questions ready
Beat the crowd
10 questions No timer
50 free Hard Multiplication trivia questions with answers — science & nature quiz, new questions added Aug 2026.
This is the hard multiplication quiz. If your times tables are automatic and you know that Napier's bones and Karatsuba's algorithm exist, these 50 questions push further: the product that reveals a hidden factorisation of 1001, the number that turns into 999999 when multiplied by 7, why 27 × 33 can be done in your head as 900 minus 9, how many multiplications Strassen needs for two 2×2 matrices, and which engineer invented a rapid-calculation system inside a concentration camp. The set mixes real products you must work out or recognise with the history and computer science of multiplying: mental shortcuts like difference of squares and casting out nines, special numbers such as 1729, 5040 and 6174, calculating tools from the soroban to Genaille–Lucas rulers, and the algorithms behind Booth, Toom–Cook, Wallace trees, Coppersmith–Winograd and AlphaTensor. It is written for people who like arithmetic to hurt a little. Every answer was checked against a primary or encyclopedic source before publishing and each question carries its citation. For a warm-up, start with our main multiplication trivia quiz and come back for this one.
30 of 50 questions with answers and explanations. Play the quiz
Q 01Which product of three consecutive primes gives 1001, the trick behind the 7, 11 and 13 divisibility test?
7 × 11 × 13
That is why splitting a number into three-digit blocks and alternately adding and subtracting them tests all three primes at once. 1001 is also the first four-digit palindrome.
Q 02Multiplying 142857 by which single digit gives 999999?
7
142857 is the repeating block of 1/7 and the best-known cyclic number: multiplying it by 1 to 6 just rotates its digits.
Q 03Using the difference-of-two-squares shortcut, 27 × 33 equals what?
891
Both numbers sit 3 away from 30, so the product is 30² minus 3², or 900 minus 9. The trick works whenever two numbers have an easily squared average.
Q 04Which four-digit number, when multiplied by 9, gives its own digits reversed?
1089
1089 × 9 = 9801. It is also 33 squared and the number that ends the classic 'reverse and subtract, then reverse and add' parlour trick.
Q 05Plato praised 5040 as an ideal city population because it is divisible by every number from 1 to 12 except which?
11
It is also 10 × 9 × 8 × 7, the number of ways to arrange 4 items from 10, and one less than the square 71².
Q 066174 is the fixed point of a digit-sorting routine devised by which Indian mathematician?
D. R. Kaprekar
Any four-digit number with at least two distinct digits reaches 6174 within seven iterations of sorting the digits, subtracting and repeating.
Q 071729 is the smallest number that is the sum of two positive cubes in two ways. By what name is it known?
The Hardy–Ramanujan
Hardy called his taxi's number dull; Ramanujan replied that it was very interesting. It is also a Carmichael number and the first nontrivial taxicab number.
Q 08144 is the only nontrivial perfect square that also appears in which famous list of numbers?
The Fibonacci sequence
Twelve dozen is a gross, and 144 is the twelfth Fibonacci number as well as 12 squared.
Q 094096 can be written as 64², 16³, 8⁴, 4⁶ and 2 to which power?
12
It is the smallest number with exactly 13 divisors and a superperfect number.
Q 102 to the power 20 equals what?
1,048,576
That is why a mebibyte is 1,048,576 bytes. It is also one of only three powers of two whose digits are all distinct, along with 2⁰ to 2¹⁵ and 2²⁹.
Q 11Which is the largest factorial that fits in a 64-bit integer?
20!
12! is the largest that fits in 32 bits. Floating point can hold bigger factorials, but only approximately.
Q 1265536 is 2 raised to which power?
16
It is the smallest number with exactly 17 divisors, and 65536 is the number of code points in a 16-bit character space.
Q 13Who coined the term 'repunit' for numbers like 11 and 111 in the 1966 book Recreations in the Theory of Numbers?
Albert H. Beiler
The word stands for 'repeated unit'. Repunits are prime for n = 2, 19, 23, 317 and 1031, among a handful of known cases.
Q 21Which Australian computer scientist devised the fast hardware multiplier tree named after him in 1964?
Chris Wallace
A Wallace tree adds up partial products in parallel using layers of adders, cutting the delay of a hardware multiplier.
Q 22Strassen's algorithm multiplies two 2×2 matrices using how many scalar multiplications instead of eight?
7
The saving compounds recursively for big matrices, at the cost of numerical stability and extra memory for seven auxiliary matrices.
Q 23Which 1990 algorithm held the fastest matrix multiplication record until 2010?
Coppersmith–Winograd
The best-known exponent for matrix multiplication now sits below 2.3714, still far from the conjectured 2.
Q 14Casting out nines checks a multiplication using what single-digit value of each number?
Its remainder on division by nine
The check works because the remainder of a product must equal the remainder of the product of the remainders. Aryabhata II described it around 950 and it appears in Liber Abaci.
Q 15Which Persian polymath, around 1020, described casting out nines as the 'Hindu method' of checking arithmetic?
Ibn Sina (Avicenna)
The earliest surviving description is in the Mahâsiddhânta of the Indian astronomer Aryabhata II, written around 950.
Q 16Jakow Trachtenberg developed his rapid mental-calculation system while held where?
In a Nazi concentration camp
The Ukrainian-Jewish engineer used the rules to keep his mind occupied. Doubleday published The Trachtenberg Speed System of Basic Mathematics in English in 1960.
Q 17The 1965 book Vedic Mathematics presents its calculation tricks as how many aphorisms and sub-aphorisms?
Sixteen sutras and thirteen sub-sutras
Scholars note the techniques have practically nothing to do with Vedic-era mathematics; several rely on decimals, which reached India only in the 16th century.
Q 18Andrew Booth invented his 1950 multiplication algorithm while researching what at Birkbeck?
Crystallography
Booth's algorithm multiplies two's-complement numbers and remains a staple of computer-architecture courses.
Q 19Toom-3 multiplication reduces the nine sub-multiplications of a three-part split to how many?
5
Andrei Toom introduced the algorithm and Stephen Cook cleaned up its description. Karatsuba is essentially the two-part case.
Q 20Karatsuba's 1960 method multiplies two two-digit numbers using how many multiplications instead of four?
3
Done recursively it beats the schoolbook method's quadratic time and set off decades of research into faster multiplication.
Q 24DeepMind's AlphaTensor searched for matrix multiplication algorithms by treating the problem as what?
A single-player game called TensorGame
It built on the reinforcement-learning approach of AlphaZero, and the discovered algorithms were released on GitHub.
Q 25Before logarithms in 1614, which trigonometry-based method was the only general way to approximate products quickly?
Prosthaphaeresis
Astronomers such as Tycho Brahe's circle used product-to-sum identities and sine tables. Its contributors included Wittich, Bürgi, Clavius and Viète.
Q 26Genaille–Lucas rulers, presented in 1891, improved on Napier's bones by doing what?
Representing the carry graphically so results are read directly
French railway engineer Henri Genaille designed them after Édouard Lucas posed a problem at the Académie in 1885. Mechanical calculators soon made them obsolete.
Q 27The Korean finger-counting method chisanbop represents numbers from 0 to what on two hands?
99
Sung Jin Pai created it in the 1940s and his son Hang Young Pai brought it to the United States in 1977. It supports multiplication and division as well.
Q 28On a Japanese soroban, each rod carries how many one-beads below the reckoning bar?
4
A single five-bead sits above the bar, giving a bi-quinary system in which every rod shows one decimal digit.
Q 29The × multiplication sign first appears in a 1618 appendix to whose book on logarithms?
John Napier
The appendix is attributed to William Oughtred, who used the same symbol in his 1631 Clavis Mathematicae. Unicode encodes it as U+00D7.
Q 30Giuseppe Peano's axioms for arithmetic define multiplication with how many axioms?
2
One handles multiplying by zero, the other by a successor. Associativity and commutativity are then proved from the axioms plus induction.