Oulun Lyseo / Galois-club
Mathemagic 1
Lightning-calculation of cubic roots of perfect cubes up to one million
HOW TO DO IT?
a) Memorize the cubes of 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 (the two first columns below). Notice
that the last digit d of the cube is respectively 1, 8, 7, 4, 5, 6, 3, 2, 9.
b) Ask spectators to choose any positive whole number n < 100 and cube it.
c) Ask the spectator to give you the result: e.g. 𝑛3 = 54872.
d) Identify the last digit (2) and conclude that the last digit of the root must be 8.
e) Drop the three last digits (872) and compare the remaining number (54) with the
values memorized in (a). It is between the cubes of 3 and 4. Conclude that the first digit
of the root must be 3.
f) Announce the answer (38).
WHY IT WORKS?
a) Let the number chosen by the spectator be 𝑛 = 10𝑡 + 𝑢 where t and u are any of the
digits 0, 1, . . . , 9. You should be able to ”guess” these digits from the value of 𝑛3 .
b) Now 𝑛3 = 𝑡 3 ∙ 1000 + 3𝑡 2 𝑢 ∙ 100 + 3𝑡𝑢2 ∙ 10 + 𝑢3 . This value is announced to you.
c) First notice that the last digit of 𝑛3 = last digit of 𝑢3 from which you can infer u using
the memorized cubes of digits.
d) Secondly, to detect t, notice that the inequality 𝑡 3 ∙ 1000 ≤ 𝑛3 < (𝑡 + 1)3 ∙ 1000 holds
true. The leftmost one is obvious. To see that the rightmost also holds, just compare the
expansion (𝑡 + 1)3 ∙ 1000 = 𝑡 3 ∙ 1000 + 3𝑡 2 ∙ 10 ∙ 100 + 3𝑡 ∙ 100 ∙ 10 + 1000 with that
of 𝑛3 above. Hence the number m of 1000’s in 𝑛3 (i.e. the number remaining when you
drop the three last digits of 𝑛3 ) falls in the interval 𝑡 3 ≤ 𝑚 < (𝑡 + 1)3. This proves that
the step e) above gives you the correct value for the first digit t of n.
e) Alternatively, instead of the algebraic argument in d), you can simply look at the two
last colums of the table below to see that the method produces the digit t correctly.
TABLE TO BE MEMORIZED
u 𝒖𝟑
Last digits
in u and 𝒖𝟑
1 1
1 and 1
2 8
2 and 8
3 27
3 and 7
4 64
4 and 4
5 125 5 and 5
6 216 6 and 6
7 343 7 and 3
8 512 8 and 2
9 729 9 and 9
10t
(𝟏𝟎𝒕)𝟑
10
20
30
40
50
60
70
80
90
1 000
8 000
27 000
64 000
125 000
216 000
343 000
512 000
729 000