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April's reverse square date

By Aziz Inan
On March 31, 2011

Since 1967, April 2nd has been celebrated as International Children's Book Day (ICBD) to inspire a love of reading and draw attention to children's books. This date was chosen to commemorate the birthday of Hans Christian Andersen (April 2, 1805-August 4, 1875), one of the most well-known writers of children's literature.

This year's ICBD will be special for another reason. Saturday, April 2nd, 2011 is a "reverse perfect square date," a term first coined last year for December 12, 2010 because the date number of this day was a reverse perfect square date. What is a reverse perfect square date?

The full date number of December 12, 2010 can be expressed in a single number as 12-12-2010, or simply, 12122010. If one reverses this number to 01022121, it is a perfect square equaling the square of 1011! This is unique because full date numbers having perfect square reverses seldom occur.

The reverse of date number 422011 corresponding to April 2nd, 2011 is 110224 which is also a perfect square, 110224 = 332 x 332! April 2nd is the third of the total 17 reverse perfect square dates to occur in this century. Notice that date number 422011 is expressed with only six digits.

In general, some full date numbers in four-digit years including all the digits of the year can be expressed as six-, seven-, or eight-digit numbers depending on the day and the month numbers. April 2nd, 2011 will be the first of the six six-digit reverse perfect square dates to occur in the 21st century. The other five will be September 4, 2013 (310249 = 557 x 557), May 2, 2045 (540225 = 735 x 735), January 4, 2054 (450241 = 671 x 671), May 2, 2079 (970225 = 985 x 985) and September 4, 2084 (480249 = 693 x 693).

April 2nd was also a reverse perfect square date in 1966 (since 669124 = 818 x 818) and will again be in 2296 (692224 = 832 x 832). Six six-digit reverse perfect square dates occurred in the 19th and 20th century each and five and seven are to occur in the 22nd and 23rd centuries respectively.

There also exist seven- and eight-digit reverse perfect square dates. Indeed, December 12, 2010's date written as 12122010 was the first of three eight-digit reverse perfect square dates contained in this century. The other two are December 14, 2030 (03024121 = 1739 x 1739) and October 20, 2062 (26020201 = 5101 x 5101). Two eight-digit reverse perfect square dates occurred in the 19th century and four in the 20th. The 22nd and 23rd centuries contain two each.

There are eight seven-digit reverse perfect square dates in this century. The first occurred on May 22, 2002 (2002225 = 1415 x 1415) and simply went unnoticed. The other seven will be May 22, 2017 (7102225 = 2665 x 2665), April 27, 2052 (2502724 = 1582 x 1582), January 21 and December 1, 2057 (7502121 = 2739 x 2739), April 4, 2063 (3602404 = 1898 x 1898), and January 29 and December 9, 2076 (6702921 = 2589 x 2589) respectively. The 19th and 20th centuries had a total of six and ten seven-digit reverse perfect square dates and 22nd and 23rd centuries will have fourteen and nine respectively.

Reverse perfect square dates are indeed fascinating because of their baffling cryptic characteristics, which is difficult to observe and track down. My hope is this article will help raise awareness of the existence of these dates and recognize their interesting hidden property. I suggest we not only enjoy this coming Saturday, April 2nd in itself but also share this story with others and especially children to turn this year's ICBD into a special one.

Aziz Inan is a professor of electrical engineering at University of Portland celebrating his 22nd year of teaching. He can be reached at 503-943-7429 or

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