Other Squares

Those poor unfortunates who had me persecute them through Year 8 may well still remember their squares to 30², even in our modern society that remembers so very little simply because it is so very easy to (re)discover information. Which of course assumes that the information found is accurate. This makes the checking process ever more important and one of the many mantras in the classroom was that one should 'check for sense'. This might involve taking your answer and putting it back into the question, or to take a different approach to the question. Many modern maths problems relate to the real world, so the 'last' part of a problem is to take your answer and in some sense interpret it for the imaginary customer. we have as a separate problem that when information is so readily available no-one has good reason to remember anything at all; try giving a colleague a phone number orally and count the repetitions you are required to make. Make a game of it and see if you can quote back even a six-digit number without writing it; so could you do that for a ten or eleven digit mobile number, even given that you 'know' it starts 079?

Enough: let's look at how we might calculate other squares at speed.

I dealt with numbers ending in 5 on a previous page. Example: 65² = 3025.

But 56² can be almost as quickly found. It ends in a 36 and the two digits in front are 25+6=31. 56²=3136.   This works: 54²=2916, 59²=3481. 

76² can be found by using (7x10+6)² = 4900 + 2x42x10 +36 = 4900 + 840 + 36. I say that it is the middle term that is the problem, the two digits multiplied together, doubled and somehow shoved in the middle. With a pencil, this would be quick enough:

76² =  4 9    3 6      87²  =   6 4   4 9          39² = 0 9  8 1         68² =  3 6  6 4
              8   4                      1 1   2                         5  4                         9  6     
          5 7   7 6                    7 5   6 9                   1 5  2 1                   4 6  2 4 

To do this with less writing, work from the right and insert 'carry' figures:

76² =   57⁸7³6,    87² = 75¹¹6⁴9,   39²=15⁶2⁸1,  68 ²= 46¹⁰2⁶4, 84²= 7 0⁶5¹6

With a little practice, these can even be done from left to right, provided we stick to two-digit numbers. Personally, such numbers would be written results, so a small amount of written calculation is justified. I cannot imagine doing this entirely orally, given the inability of most of our population to remember any number of length.


Can we stretch this technique to three-digit numbers? We can and here is the theory: (100a+10b+c)² = 10⁴a² + 10²b² + c² + 2000ab + 20bc + 200ac.

So a written technique requires three cross-products all doubled, 2ab, 2bc, 2ac.

782² = 4  9 | 6  4 | 0 4   that's a², b², c²
            1  1  2 | 3  2       112 =2x7x8=2ab, 32=2x8x2=2bc. Using | as a separator.
                    2   8             28  = 2x7x2 = 2ac. Now add them up in place
            6  1   1  5  2  4      Which is correct, 782² = 611524.   Try some yourself.


DJS 20190714

is 20190714 = 2x3xprime?   Approximate √3365119




Upper bound √3 36 51 19 < 361 00 00 =1900².  

Lower bound √3 36 51 19>324 00 00=1800². 

Linear interpolation suggests 1800+ 12.5/37x100 = 1833.7 will be too small. So 1834 is expected to be the nearest integer solution



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