Digit Deduction

Logic Level 3

I've chosen a two-digit positive integer $n$ (i.e. 10 - 99) and tell its first digit to Ojas, its second digit to Paddy, and its digit sum to Quentin. The following conversation occurs, in which all three of them have perfect deduction skills, and "only now" means since the previous piece of knowledge:

Paddy: " $n$ isn't prime."
Ojas: " $n$ isn't a perfect square."
Paddy: "I'm not sure if $n$ is a triangle number or not."
Quentin: "I know what $n$ is. But only now ."
Ojas: "I only now know what $n$ is."

What is $n?$

The answer is 95.

1 solution

Stephen Mellor
Sep 1, 2018

Start out with a 9x10 grid showing all possible $n$ .

10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
50	51	52	53	54	55	56	57	58	59
60	61	62	63	64	65	66	67	68	69
70	71	72	73	74	75	76	77	78	79
80	81	82	83	84	85	86	87	88	89
90	91	92	93	94	95	96	97	98	99

Paddy, who knows the last digit, knows for sure that $n$ isn't prime, meaning his last digit is 0,2,4,5,6 or 8.

10	12	14	15	16	18
20	22	24	25	26	28
30	32	34	35	36	38
40	42	44	45	46	48
50	52	54	55	56	58
60	62	64	65	66	68
70	72	74	75	76	78
80	82	84	85	86	88
90	92	94	95	96	98

Ojas, who knows the first digit, knows for sure that $n$ isn't a square number, so it must be in a row which doesn't have any square numbers still in it (since Ojas has already narrowed it down to these possibilities. For example, Ojas could have that the first digit is a 4, even though 49 is a square, but still say this as he knows it can't be 49 from Paddy's first statement). This means that the first digit is 4,5,7,8 or 9.




40	42	44	45	46	48
50	52	54	55	56	58

70	72	74	75	76	78
80	82	84	85	86	88

||90|| ||92|| ||94||95||96|| ||98|| ||

Paddy, who knows the last digit, isn't sure if $n$ is a triangle number or not. This means that $n$ must be in a column (last digit) where there is a triangle number. The triangle numbers remaining are 45,55 and 78 so the last digit is a 5 or 8.




					45			48
					55			58

					75			78
					85			88
					95			98

Quentin, who knows the digit sum, now says that he knows what $n$ is. This means that the digit sum must be unique so $n$ is 45,55,78,88,95 or 98. However, he only now knew it, meaning that the digit sum must not have been unique before, so it can't be 98.




					45
					55

								78
								88

|| || || || || ||95|| || || || ||

Ojas only now knows what $n$ is, which is since 98 has been ruled out, so $n$ must be 95.

10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
50	51	52	53	54	55	56	57	58	59
60	61	62	63	64	65	66	67	68	69
70	71	72	73	74	75	76	77	78	79
80	81	82	83	84	85	86	87	88	89
90	91	92	93	94	95	96	97	98	99

10	12	14	15	16	18
20	22	24	25	26	28
30	32	34	35	36	38
40	42	44	45	46	48
50	52	54	55	56	58
60	62	64	65	66	68
70	72	74	75	76	78
80	82	84	85	86	88
90	92	94	95	96	98

10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
50	51	52	53	54	55	56	57	58	59
60	61	62	63	64	65	66	67	68	69
70	71	72	73	74	75	76	77	78	79
80	81	82	83	84	85	86	87	88	89
90	91	92	93	94	95	96	97	98	99

10	12	14	15	16	18
20	22	24	25	26	28
30	32	34	35	36	38
40	42	44	45	46	48
50	52	54	55	56	58
60	62	64	65	66	68
70	72	74	75	76	78
80	82	84	85	86	88
90	92	94	95	96	98

Digit Deduction

The answer is 95.

1 solution

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10	11	12	13	14	15	16	17	18	19
20	21	22	23	24	25	26	27	28	29
30	31	32	33	34	35	36	37	38	39
40	41	42	43	44	45	46	47	48	49
50	51	52	53	54	55	56	57	58	59
60	61	62	63	64	65	66	67	68	69
70	71	72	73	74	75	76	77	78	79
80	81	82	83	84	85	86	87	88	89
90	91	92	93	94	95	96	97	98	99

10	12	14	15	16	18
20	22	24	25	26	28
30	32	34	35	36	38
40	42	44	45	46	48
50	52	54	55	56	58
60	62	64	65	66	68
70	72	74	75	76	78
80	82	84	85	86	88
90	92	94	95	96	98