Maths Tricks and Facts
SSS =
Strange Square Sums
Look at 9, 10 and 45:
- 9²
is equal to 81
- 10²
is equal to 100
- 45²
is equal to 2025
Split the squares into two pieces then add those two values:
- 8
+ 1 = 9
- 10
+ 0 = 10
- 20
+ 25 = 45
Did you notice? The sums are equal to your starting numbers.
- 9²
= 81 8 + 1 = 9
- 10²
= 100 10 + 0 = 10
- 45²
= 2025 20 + 25 = 45
How many other positive integers can you find that this is
true for?
Feel free to stop here and see how many you can find!
(scroll down)
(to see)
(some of)
(my answers)
Here are the first nine that I found:
- 9²
= 81 8 + 1 = 9
- 10²
= 100 10 + 0 = 10
- 45²
= 2025 20 + 25 = 45
- 55²
= 3025 30 + 25 = 55
- 99²
= 9801 98 + 01 = 99
- 100²
= 10000 100 + 00 = 100
- 297²
= 88209 88 + 209 = 297
- 703²
= 494209 494 + 209 = 703
- 999²
= 998001 998 + 001 = 999
Here are some more:
- 1000²
= 1000000 1000 + 000 = 1000
- 2223²
= 4941729 494 + 1729 = 2223
- 2728²
= 7441984 744 + 1984 = 2728
- 4950²
= 24502500 2450 + 2500 = 4950
- 5050²
= 25502500 2550 + 2500 = 5050
- 7272²
= 52881984 5288 + 1984 = 7272
- 7777²
= 60481729 6048 + 1729 = 7777
- 9999²
= 99980001 9998 + 0001 = 9999
- 10000²
= 100000000 10000 + 0000 = 10000
- 17344²
= 300814336 3008 + 14336 = 17344
- 22222²
= 493817284 4938 + 17284 = 22222
- 77778²
= 6049417284 60494 + 17284 = 77778
- 82656²
= 6832014336 68320 + 14336 = 82656
- 95121²
= 9048004641 90480 + 04641 = 95121
- 99999²
= 9999800001 99998 + 00001 = 99999
- 9²
= 81 → 8 + 1 = 9
- 10²
= 100 → 10 + 0 = 10
- 45²
= 2025 → 20 + 25 = 45
- 55²
= 3025 → 30 + 25 = 55
- 99²
= 9801 → 98 + 01 = 99
- 100²
= 10000 → 100 + 00 = 100
- 297²
= 88209 → 88 + 209 = 297
- 703²
= 494209 → 494 + 209 = 703
- 999²
= 998001 → 998 + 001 = 999
- 1000²
= 1000000 → 1000 + 000 = 1000
- 2223²
= 4941729 → 494 + 1729 = 2223
- 2728²
= 7441984 → 744 + 1984 = 2728
- 4950²
= 24502500 → 2450 + 2500 = 4950
- 5050²
= 25502500 → 2550 + 2500 = 5050
- 7272²
= 52881984 → 5288 + 1984 = 7272
- 7777²
= 60481729 → 6048 + 1729 = 7777
- 9999²
= 99980001 → 9998 + 0001 = 9999
- 10000²
= 100000000 → 10000 + 0000 = 10000
- 17344²
= 300814336 → 3008 + 14336 = 17344
- 22222²
= 493817284 → 4938 + 17284 = 22222
- 77778²
= 6049417284 → 60494 + 17284 = 77778
- 82656²
= 6832014336 → 68320 + 14336 = 82656
- 95121²
= 9048004641 → 90480 + 04641 = 95121
- 99999²
= 9999800001 → 99998 + 00001 = 99999
- 100000²
= 10000000000 → 100000 + 00000 = 100000
- 142857²
= 20408122449 → 20408 + 122449 = 142857
- 148149²
= 21948126201 → 21948 + 126201 = 148149
- 181819²
= 33058148761 → 33058 + 148761 = 181819
- 187110²
= 35010152100 → 35010 + 152100 = 187110
- 208495²
= 43470165025 → 43470 + 165025 = 208495
- 318682²
= 101558217124 → 101558 + 217124 = 318682
- 329967²
= 108878221089 → 108878 + 221089 = 329967
- 351352²
= 123448227904 → 123448 + 227904 = 351352
- 356643²
= 127194229449 → 127194 + 229449 = 356643
- 390313²
= 152344237969 → 152344 + 237969 = 390313
- 461539²
= 213018248521 → 213018 + 248521 = 461539
- 466830²
= 217930248900 → 217930 + 248900 = 466830
- 499500²
= 249500250000 → 249500 + 250000 = 499500
- 500500²
= 250500250000 → 250500 + 250000 = 500500
- 533170²
= 284270248900 → 284270 + 248900 = 533170
- 538461²
= 289940248521 → 289940 + 248521 = 538461
- 609687²
= 371718237969 → 371718 + 237969 = 609687
- 643357²
= 413908229449 → 413908 + 229449 = 643357
- 648648²
= 420744227904 → 420744 + 227904 = 648648
- 670033²
= 448944221089 → 448944 + 221089 = 670033
- 681318²
= 464194217124 → 464194 + 217124 = 681318
- 791505²
= 626480165025 → 626480 + 165025 = 791505
- 812890²
= 660790152100 → 660790 + 152100 = 812890
- 818181²
= 669420148761 → 669420 + 148761 = 818181
- 851851²
= 725650126201 → 725650 + 126201 = 851851
- 857143²
= 734694122449 → 734694 + 122449 = 857143
- 961038²
= 923594037444 → 923594 + 037444 = 961038
- 994708²
= 989444005264 → 989444 + 005264 = 994708
- 999999²
= 999998000001 → 999998 + 000001 = 999999
- 1000000²
= 1000000000000 → 1000000 + 000000 = 1000000
- 4444444²
= 19753082469136 → 1975308 + 2469136 = 4444444
- 4927941²
= 24284602499481 → 2428460 + 2499481 = 4927941
- 5072059²
= 25725782499481 → 2572578 + 2499481 = 5072059
- 5555556²
= 30864202469136 → 3086420 + 2469136 = 5555556
- 9372385²
= 87841600588225 → 8784160 + 0588225 = 9372385
- 9999999²
= 99999980000001 → 9999998 + 0000001 = 9999999
- 10000000²
= 100000000000000 → 10000000 + 0000000 = 10000000
- 11111112²
= 123456809876544 → 1234568 + 09876544 = 11111112
- 13641364²
= 186086811780496 → 1860868 + 11780496 = 13641364
- 16590564²
= 275246813838096 → 2752468 + 13838096 = 16590564
- 19273023²
= 371449415558529 → 3714494 + 15558529 = 19273023
- 19773073²
= 390974415863329 → 3909744 + 15863329 = 19773073
- 24752475²
= 612685018625625 → 6126850 + 18625625 = 24752475
- 25252525²
= 637690018875625 → 6376900 + 18875625 = 25252525
- 30884184²
= 953832821345856 → 9538328 + 21345856 = 30884184
- 36363636²
= 1322314023140496 → 13223140 + 23140496 = 36363636
- 38883889²
= 1511956823764321 → 15119568 + 23764321 = 38883889
- 44363341²
= 1968106024682281 → 19681060 + 24682281 = 44363341
- 44525548²
= 1982524424700304 → 19825244 + 24700304 = 44525548
- 49995000²
= 2499500025000000 → 24995000 + 25000000 = 49995000
- 50005000²
= 2500500025000000 → 25005000 + 25000000 = 50005000
- 55474452²
= 3077414824700304 → 30774148 + 24700304 = 55474452
- 55636659²
= 3095437824682281 → 30954378 + 24682281 = 55636659
- 61116111²
= 3735179023764321 → 37351790 + 23764321 = 61116111
- 63636364²
= 4049586823140496 → 40495868 + 23140496 = 63636364
- 69115816²
= 4776996021345856 → 47769960 + 21345856 = 69115816
- 74747475²
= 5587185018875625 → 55871850 + 18875625 = 74747475
- 75247525²
= 5662190018625625 → 56621900 + 18625625 = 75247525
- 80226927²
= 6436359815863329 → 64363598 + 15863329 = 80226927
- 80726977²
= 6516844815558529 → 65168448 + 15558529 = 80726977
- 83409436²
= 6957134013838096 → 69571340 + 13838096 = 83409436
- 86358636²
= 7457814011780496 → 74578140 + 11780496 = 86358636
- 88888888²
= 7901234409876544 → 79012344 + 09876544 = 88888888
- 91838088²
= 8434234407495744 → 84342344 + 07495744 = 91838088
- 94520547²
= 8934133805179209 → 89341338 + 05179209 = 94520547
- 99999999²
= 9999999800000001 → 99999998 + 00000001 = 99999999
- 100000000²
= 10000000000000000 → 100000000 + 00000000 = 100000000
- 234567901²
= 55022100179545801 → 55022100 + 179545801 = 234567901
- 332999667²
= 110888778222110889 → 110888778 + 222110889 = 332999667
- 432432432²
= 186997808245434624 → 186997808 + 245434624 = 432432432
- 567567568²
= 322132944245434624 → 322132944 + 245434624 = 567567568
- 667000333²
= 444889444222110889 → 444889444 + 222110889 = 667000333
- 765432099²
= 585886298179545801 → 585886298 + 179545801 = 765432099
- 999999999²
= 999999998000000001 → 999999998 + 000000001 = 999999999
- 1000000000²
= 1000000000000000000 → 1000000000 + 000000000 = 1000000000
- 1111111111²
= 1234567900987654321 → 123456790 + 0987654321 = 1111111111
- 1776299581²
= 3155240201460775561 → 315524020 + 1460775561 = 1776299581
- 2020202020²
= 4081216201612080400 → 408121620 + 1612080400 = 2020202020
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