Square and Cube numbers memory techniques

[chaitanya10 (CHAITANYA DASARI)]

Hi,

I have seen one memory athlete demonstrating square numbers and cube numbers. When he as about square of 87 and 76 he was able to tell in a jiffy. Also when he was given last 3 digits of a cube number, he completed remaining 3 digits and he told its cube root immediately. What memory techniques he could have used to learn these concepts.

[bjoern.gumboldt]

@chaitanya10

What memory techniques he could have used to learn these concepts.

That depends… you can either use memory techniques or simply concepts from mental math.

@chaitanya10

When he as about square of 87 and 76 he was able to tell in a jiffy.

Unfortunately, “a jiffy” is not a very quantitative way of referring to things… let’s take the mental math approach first for the 76^2 above.

Squares ending in 0 or 5 are super easy to calculate. Pick 75; since it’s closer than 80. To calculate a two digit number a|b (in this case 7|5), multiply a(a+1) for the left hand side (here 7*8) and square b for the right hand side (always 25). To get from 75^2=5,625 to 76^2, you just double the number you squared (here 75*2=150 and add it on top, plus an additional 1 for 5,625+150+1=5,776

For a bit more in-depth about this approach have a look here…

How to calculate squares quickly

It’s a shame we only have a like button, because this post is 5 out of 5 stars! The only way this could be better (five-and-a-half stars) is if you’d have called if squares with binomial formulas. It’s pretty clear that you are using the first and second formula when adding and subtracting, but even the numbers ending in 5 are done with the third: \color{blue}{(a+b)^2=a^2+2ab+b^2} \color{blue}{(a-b)^2=a^2-2ab+b^2} \color{blue}{(a+b)*(a-b)=a^2-b^2} For those interested, the nice thing he…

@chaitanya10

Also when he was given last 3 digits of a cube number, he completed remaining 3 digits and he told its cube root immediately.

I don’t understand what you mean by “completed remaining 3 digits” but cube roots are easier than square roots. Just search here on the forum or google “Vedic math cube roots”. Both your question have been explained before on here, so you shouldn’t have a problem finding the information using the search function.

[chaitanya10 (CHAITANYA DASARI)]

Thank you so much for your time and inputs.
Cube roots -

@bjoern.gumboldt

I don’t understand what you mean by “completed remaining 3 digits” but cube roots are easier than square roots. Just search here on the forum or google “Vedic math cube roots”. Both your question have been explained before on here, so you shouldn’t have a problem finding the information using the search function.

When the memory trainer was given the number 616, he said it is part of 175616 whose cube root is 56. He did this activity with blindfold.

[Daniel_360 (Daniel Timms)]

For 2-digit squares and cubes, it’s pretty easy to memorize these directly using flashcards (e.g. spaced repetition software on cram mode) I’ve done it for squares, but not for cubes. I had an Indian kid in a workshop once who knew the cubes by heart.

These days I use more advanced tool that I’m still experimenting with, but the principle is the same.

For squares, you should see some patterns in the data. For example, notice the following:

• 23² = 529
• 27² = 729
• 73² = 5329
• 77² = 5929

The squares of X, 50 – X, 50 + X and 100 – X all end in the same two digits.

You can also see that the squares above 90 all have an even hundreds digit, and end in the single-digit squares 81, 64, etc.:

• 8281
• 8464
• 8649
  etc.

You’ll find many more patterns if you write out the examples and study them.

[bjoern.gumboldt]

@Daniel_360

For squares, you should see some patterns in the data.

@chaitanya10 you can have a look here for the symmetry around 25.

How can I start mental calculating

Making progress with your squares, @Badr4sudan? I mentioned a few more shortcuts in my last reply, so here’s one. Let’s say you’ve made it up to 25² using the methods I’ve mentioned previously. There is a nice symmetry at 25 once you get there: [25-50] Use 25 as your base… Δ is the difference to 25. Say you want to square 28 in which case Δ = (28 - 25) = 3; so just take the square of (25 - Δ) = 22² = 484 and add to it (Δ x 100) = 300. The result is 484 + 300 = 784 Example: 32² Δ = 7 → …

…and once you’re at 50, you got the symmetry that @Daniel_360 already mentioned. That let’s you technically go all the way up to 100; however, I’d use the symmetry around 75 instead to make the calculation easier.

@Daniel_360

it’s pretty easy to memorize these directly using flashcards

Alternatively, you could build your PAO system in such a way that the O is a 3-digit number that represents the square of P. In mine, for example, I have “Kylie Minogue squeezing a LeMoN”

KM is 73 with a square of LMN which is 532_ and then you just append the unit digit squared (i.e., 3^2=9) to get 73^2=5,329 and that’d how you get the squares up to 100 without even calculating.

[bjoern.gumboldt]

@chaitanya10

When the memory trainer was given the number 616, he said it is part of 175616 whose cube root is 56.

I still don’t get it… “is part of”… what does that mean? 616 could be part of any number… you guys could have given him 615 or 614 instead… then what… there’s no perfect cube ending in these digits.

Did he ask you guys to use a calculator to cube a 2-digit number and then just give him the last 3 digits for him to complete?

@chaitanya10

He did this activity with blindfold.

About as impressive as listening to the radio blindfolded. I’m sure he could have tied his hands behind his back too. How does either make it more challenging to do cubes or squares? It’s one thing to do a Rubik’s cube blindfolded but here I really don’t see the point… total gimmick.

EDIT: just watched the video below. This was done during an online presentation to ensure the other participants that he is not reading off the result from somewhere rather than an obstacle to make the exercise harder. All good then!

@chaitanya10

175616 whose cube root is 56.

1. Learn the single digit cubes 0 through 9: 0, 1, 8, 27, 64, 125, 216, 343, 512, 729

2. Cubes end in the same digits as the number cubed. Exceptions are 8/2 and 7/3 where it’s flipped: 8 is 2 and 2 is 8; 7 is 3 and 3 is 7.

3. Separate the last 3 digits…

   175 | 616

4. Find the biggest cube smaller or equal to 175 (see 1.), which is 5^3=125

5. Append to 5_ the corresponding unit digit for 616, which is _6 for 56

778 | 688

4. 729 is less than 778 from 1. above for 9_
5. 688 end in 8 which is _2 according to 2. above, so you get 92

132 | 651

4. 125 gets you 5
5. append 1 to get 51

125 | 000

4. 5_ ← 125
5. _0 → 50

117 | 649

4. 4_ ← 64
5. _9 → 49

Hope that makes sense… you might not know why… but you should know how now.

[chaitanya10 (CHAITANYA DASARI)]

@bjoern.gumboldt

Alternatively, you could build your PAO system in such a way that the O is a 3-digit number that represents the square of P. In mine, for example, I have “Kylie Minogue squeezing a LeMoN”

KM is 73 with a square of LMN which is 532_ and then you just append the unit digit squared (i.e., 3^2=9 32=93^2=9 ) to get 73^2=5,329 732=5,32973^2=5,329 and that’d how you get the squares up to 100 without even calculating.

Thank you sir - Sounds interesting

[magicpauluk (Paul Andrews)]

I do this for calculating the square and cube of numbers with a Person / Object type method.

I have a set cast of characters that I always use for the number 00 to 99

Then I created words for the calculation results. Using a number to letter system. (I use Rhyming Letter Getters, but works same with Dominic System or Major System)

So for example 41 in my cast is David Beckham, the square of 41 is 1681 which translates as BSGB (Using the number to letter code that I use)

Then I inserted ‘Null letters’ and created the words “Bus Globe”. Whilst a Bus Globe is not a real thing it can certainly be visualised.

So I mentally created a movie scene that links David Beckham to Bus Globe and I have now linked 41 with 1681 in a visual movie.

Using the Dominic System 41 would be DA, so that could be David Attenborough - 1681 would be ASHA - I could insert ‘null letters’ and make it AStHmA - so imagining David Attenborough with Asthma I’d have a scene that would remind me of the answer to the sum.

[bjoern.gumboldt]

@magicpauluk

I do this for calculating the square and cube of numbers with a Person / Object type method.

Are you saying that’s how you do do it or how you would do it?

@magicpauluk

So for example 41 in my cast is David Beckham, the square of 41 is 1681 which translates as BSGB (Using the number to letter code that I use)

41^3=68,921 though and now you’re translating 5 letters? And once you get past 46, it’s 6 letter translations? Isn’t it kinda difficult to find words then?

[magicpauluk (Paul Andrews)]

I do this for Square numbers but only as a party piece. Creating the phrases was pretty easy but the words are quite obscure. But it’s a fixed list - so I only ever had to do it once.

For longer numbers (sums, codes or references) I use a Person to Person/Action/Object/Quote type method which is really effective.

I link my cast member for the number with the POAQ attributes of the answer (67 cubed - 67 is 300,773) - 67 in my cast is Stephen Hawking and then the result split into pairs gives me 30-07-63 - These relate to 3 cast members, and I’d create an image of Stephen Hawing interacting with them in normal person, action object way.

[fred2 (fred)]

Just sticking with the “Squares” of numbers, that sounds well and good in theory but in practice?? For example, my 99 is Nick Nolte (as I am using the Dominic System). NN squared = 9801 or NHOA. Now here’s the rub, how can I get anything for 99 (NN) from these letters when the only vowels I can use that are not already being catered for in Dominic system are the: I and U (vowels) as in Dominic system: A=1, E=5 and O=0. I have always thought the Dominic system as superior to the Major System but from the above example 99 in Major could translate into PP (Peter Pan). 9801 in Major System translates into PFST. Pea FeaST or even BFST (BeeF STew). I am beginning to believe well at least for “squaring” and “Cubing” of two digit numbers that the Major System trumps the Dominic System as you have the advantage of being able to use every vowel as a ‘silent filler’!

[fred2 (fred)]

Or perhaps to think outside the box a little, one could use both Dominic System and Major Systems alongside one another. For example, Dominic’s NN or 99 (Nick Nolte) could be enjoying a BeeF STew = 9801). Dominic’s DA = 41 (David Attenborough) could be eating Hollandse Nieuwe Haring which in English is “Scoused Herrings” or Dutch Food (1681). I can picture David Attenborough stuffing his face full with “Scoused Herrings”, hence 41 squared = Dutch Food = 1681. In fact David Attenborough has stuffed himself so full with “Scoused Herrings” (Dutch Food) that he is having an AStHmA attack bringing it full circle back to Dominic system again. LOL!

[magicpauluk (Paul Andrews)]

Hi Fred, I too thought the Dominic System was above the Major System. But didn’t like the fact it used vowels. Now I only ever use Rhyming Letter Getters because this system also leaves all the vowels as ‘null’ letters (silent fillers). Coincidentally I also have Nick Nolte in my cast list (for 00 as that becomes NN using RLG’s). 9801 translates at WGNB - inserting null letters I get “WaGoN Bar”.

I think mixing systems would be useful for some aspects, but I do prefer just have definitive letters for each number.