2 X 2 = 3

If simple multiplication, TWO x TWO = THREE, where each letter stands for a different digit, find WHO. Note: Just as TWO is a three digit number, THREE is a 5-digit number.


The answer is 398.

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1 solution

Chew-Seong Cheong
Feb 15, 2015

We are given that T W O 2 = T H R E E \overline{TWO}^2 = \overline{THREE} .

This tells us that T W O < 100000 = 316.2 \overline{TWO} < \sqrt{100000} = 316.2 , therefore T T is 1 1 , 2 2 or 3 3 , It is obvious that T = 1 T = \boxed{1} because:

  • if T = 2 T=2 then the T T of T H R E E \overline{THREE} will be either 4 4 or 5 T 5 \ne T
  • if T = 3 T=3 then the T T of T H R E E \overline{THREE} will be 9 T 9 \ne T

Therefore, T H R E E < 20000 T W O < 20000 = 141.4 \overline{THREE} < 20000 \quad \Rightarrow \overline{TWO} < \sqrt{20000} = 141.4

We also note that for the last digits O × O = 10 c 1 + E O \times O = 10c_1 + E , where c 1 = 0 , 1 , 2 , 3 , 4 , 6 , 8 c_1 = 0,1,2,3,4,6,8 is the carried-forward value, O 0 , 1 , 5 , 6 O \ne 0,1,5,6 else O = E O=E which is unacceptable.

Working out the possible cases as follows:

\[\begin{array} {} 124^2=15376\\ 127^2=16129\\ 128^2=16384\\ 129^2=16641\\ 132^2=17424\\ 134^2=17956\\ 137^2=18769\\ 138^2=19044\\ 139^2=19321 \end{array}\]

We note that there is only one solution: T W O = 138 \overline{TWO} = 138 and T H R E E = 19044 \overline{THREE} = 19044

W H O = 398 \Rightarrow \overline{WHO} = \boxed{398}

Used the same method :) Also it may be important to explain why W {W} = 2 or 3

Curtis Clement - 6 years, 3 months ago

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I believe that it becomes clear that this must be true when he states that T W O < 142 \overline{TWO} < 142 ...

Alexandre Miquilino - 6 years, 3 months ago

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