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Doing Math In My Head, (instead of sleeping) Again.

It occurred to me that for every square, say 30 x 30, you can get to the next one, 31 x 31 by simply adding the first and the second numbers on.  So 31 squared is 30 squared + 30 +31.  It becomes more obvious when you think of a grid:  30 rows x 30 columns will go to 31 rows and 31 columns by adding a row of 30 and then a column of 31.

It thus follows that if any two consecutive numbers add to a perfect square, then the three could form a right triangle of whole numbers.  Example:  12 + 13 = 25 which is 5 squared, so 12, 13 and 5 will make up the sides.  It must be the case, since you get from 12 squared (144) to 13 squared (169) by adding 5 squared (25).

Just in my head, I found one which would have been too high to calculate:  112 + 113 = 225 and the square root of 225 is 15.

I wonder if there's a name for this pattern of consecutive numbers and then a third number, which together make a perfect square.

I asked GPT about this and while there doesn't seem to be a name for this, it was known in ancient times.  It works for any odd square because there will always be two consecutive numbers which add to this value.  It obviously cannot work for even squares, since you can't add consecutive integers and reach an even number.

You can go as high as you want! 6,160, 111, 6,161 works, but you may want to check with a calculator!

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Allow Me to Rebut...

 Me in Bold

Arnaud Bertrand

@RnaudBertrand

If Europeans were paying attention (or being told the truth), they should be beyond appalled by this "deal": https://bbc.com/news/articles/cx2xylk3d07o It's nothing more than one of the most expensive imperial tributes in history. Just a massive one-way transfer of wealth with no reciprocal benefits The "deal" is: - The EU now gets charged 15% tariffs on its exports to the US when they commit to charging zero tariffs on US imports in the EU

The EU currently charges a VAT on imports from the US that we do not charge on imports from the EU. This tariff makes up for that imbalance.

- The EU agrees to invest $600 billion in the US, for no other obvious reason than pleasing "daddy"

Investments that EU based companies would have made anyway, because of poor business opportunities in the EU and good opportunities here.

- The EU will "purchase hundreds of billions of dollars of American military equipment"

Now that the EU has realized that defense is still necessary and already agreed to the minimum spending levels of all NATO countries, they need to spend money. Most of the money will go to EU countries, but the US is a top player and if they want the best, they will need to buy some US equipment.

- The EU commits to buying 750 billion dollars worth of very expensive US LNG, specifically $250 billion for each of the next 3 years

US LNG is competitive to other suppliers of LNG and the only alternative is Russia, which would allow them to blackmail the EU with regards to Ukraine. There are good shale plays in the EU, but Europe is too stupid to make their own gas, so here we are. In exchange for all these concessions and extraction of their wealth they get... nothing. I'm not even exaggerating, that IS the deal: the EU gets nothing.

They get LNG, good military equipment that they need and a lower tariff than they would have gotten without a deal. This does not even remotely ressemble [sic] the type of agreements made by two equal sovereign powers. It rather looks like the type of unequal treaties that colonial powers used to impose in the 19th century - except this time, Europe is on the receiving end. More worryingly, this sets a dynamic and a precedent: what do you think happens next from here? In the 19th century, were colonial powers content with their first unequal treaty? Of course not - one of the key rules of geopolitics is that weakness only encourages further exploitation.

The EU chose to be weak, they can choose to be powerful again. In the meantime, weakness has a cost. Again, this is Europe's century of humiliation.

Doing Math In My Head

Just as I went to bed the other night, unbidden, I remembered reading that a few days ago, we had a date with a perfect square:  7-24-25.  At least I thought that was the date and set-about confirming it in my head.  7 x 7 = 49 was easy.  Next up was 25 x 25.  I figured 4 x 25 =100 and 4 goes into 25, 6 times with a remainder of 1, so 625.  I could do 24 x 24 the same way, but it would be 6 x (100 -4), which would be 600-24, so I'm missing 24 and the remainder of 25 for a total of 49.

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