Looking back at the Sandia Mountains

 

Late in the afternoon on October 15th we drove up to Sandia Crest, which at 10,679 ft. looms large to the northeast of Albuquerque. The previous post showed views from there. The next morning in our hotel’s parking lot I noticed that the view back toward the mountains, now covered with fog and clouds, was dramatic—at least if I could ignore light poles, buildings, billboards, highways, and other trappings of the city. To exclude as much of that as possible, for the top picture I zoomed my telephoto lens to its maximum focal length of 400mm. With a change of scale and locale you might see an ocean wave breaking near the shore. Three-quarters of an hour later and miles further north as we wended our way toward Santa Fe, the land added color to the still-shrouded mountains.

 

  

 

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Look at this:

 

1    3    5    7    9    11    13    15    17    19    21    23    25    27    29  …

 

If I asked you what it is, you’d say it’s the odd numbers.

(Or if you wanna get fancy, you’d say it’s the positive odd numbers.)

You’ve been seeing these numbers for a long time, ever since you were in elementary school.

Now here’s something you might never have noticed.

 

The “sum” of the first 1 odd number is 1, which is 1 x 1, or 1 squared.

The sum of the first 2 odd numbers is 1 + 3, or 4, which is 2 x 2, or 2 squared.

The sum of the first 3 odd numbers is 1 + 3 + 5, or 9, which is 3 x 3, or 3 squared.

The sum of the first 4 odd numbers is 1 + 3 + 5 + 7, or 16, which is 4 x 4, or 4 squared.

The sum of the first 5 odd numbers is 1 + 3 + 5 + 7 + 9, or 25, which is 5 x 5, or 5 squared.

The sum of the first 6 odd numbers is 1 + 3 + 5 + 7 + 9 + 11, or 36, which is 6 x 6, or 6 squared.

The sum of the first 7 odd numbers is 1 + 3 + 5 + 7 + 9 + 11 + 13, or 49, which is 7 x 7, or 7 squared.

 

Now if I skipped ahead 93 lines and asked you what the sum of the first 100 odd numbers is, you wouldn’t have to do any adding at all. You’d chime right in and say with verve and élan that the sum of the first 100 odd numbers is 100 x 100, or 10,000.*

  

If you’d like a nifty visual explanation for why the sum of
the first however many odd numbers is always a square, here it is:

 

 

But if you’d also like an explanation of where the word nifty came from, you’re outta luck: no one knows.

 

* One day in the early 1970s I was in a supermarket in my home town of Franklin Square, New York. As I walked down an aisle in that large store I passed a woman who was telling her young daughter that ten times ten is a hundred and a hundred times a hundred is a thousand. Why she was saying that, I don’t know; why I didn’t intervene and correct her I also don’t know. Evidence points more to politeness than a lack of boldness. That’s because in the same supermarket minutes later when I heard two women talking about pressure cookers, with each saying they had one but didn’t use it, I approached them and asked if I could have those pressure cookers that they didn’t use. I ended up several days later with two pressure cookers, one of which, a fancy stainless steel model with a copper bottom, had been a wedding present and was essentially brand new. I used it for decades until finally part of the handle came loose and the rubber gasket no longer sealed properly, and replacement parts were no longer available for such an old pressure cooker.

 

© 2022 Steven Schwartzman