1.2.9 Squares from 41-59

There is a quick trick for computing squares of numbers from 41 to 59. Any number in this range can be expressed as (50 ± k) where k is the distance from 50.

The Derivation

(50 ± k)² = 2500 ± 100·k + k²

= 100(25 ± k) + k²

The Rule

1. Tens/Ones digits = the distance from 50, squared (k²)
2. Rest of answer = 25 + k (if number > 50) or 25 − k (if number < 50)

Note: You can extend this concept to squares outside the range of 41-59 as long as you keep track of carries appropriately.

Example: 46²

Step	Calculation	Result
Distance from 50	50 − 46	4
Tens/Ones	4²	16
Rest of Answer	25 − 4	21

Answer: 2116

Example: 57²

Step	Calculation	Result
Distance from 50	57 − 50	7
Tens/Ones	7²	49
Rest of Answer	25 + 7	32

Answer: 3249

Example with Carry: 61²

When the square of k is greater than 99, you need to carry:

Step	Calculation	Result
Distance from 50	61 − 50	11
Tens/Ones	11² = 121	21 (carry 1)
Rest of Answer	25 + 11 + 1	37

Answer: 3721

Quick Reference Table

Number	k	k²	25 ± k	Answer
41	9	81	25 − 9 = 16	1681
42	8	64	25 − 8 = 17	1764
43	7	49	25 − 7 = 18	1849
44	6	36	25 − 6 = 19	1936
45	5	25	25 − 5 = 20	2025
46	4	16	25 − 4 = 21	2116
47	3	09	25 − 3 = 22	2209
48	2	04	25 − 2 = 23	2304
49	1	01	25 − 1 = 24	2401
50	0	00	25	2500
51	1	01	25 + 1 = 26	2601
52	2	04	25 + 2 = 27	2704
53	3	09	25 + 3 = 28	2809
54	4	16	25 + 4 = 29	2916
55	5	25	25 + 5 = 30	3025
56	6	36	25 + 6 = 31	3136
57	7	49	25 + 7 = 32	3249
58	8	64	25 + 8 = 33	3364
59	9	81	25 + 9 = 34	3481

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Problem Set 1.2.9

Practice these problems. Type your answer and press Enter to check:

Basic Squares (41-59)

Squares Above 50

Extended Range (with carries)

Applied Problems

Challenge Problems