You do not need to know the house number

Logic Level 3

The following is a conversation between Gabriel and Heather:

Gabriel: "I am thinking of two distinct single-digit numbers. Can you guess the sum of these two numbers?"
Heather: "No. Can you give me a clue?"

Gabriel: "The last digit of the product of the two numbers is your house number."
Heather: "Now I know the sum of the two numbers."

So, what is the sum of the two numbers?

Note: It is possible that the product of the two numbers is a single-digit.

The answer is 10.

6 solutions

Siva Budaraju
Apr 16, 2017

The only way Heather could guess the sum is if she could somehow have determined the was only 1 unique sum that was possible. The only way she could have determined the numbers (or sets) are if her house number could only be expressed as the last digit of the product of 2 numbers in exactly 1 way, or if the last digit could be expressed as multiple products that all summed to the same number.

So, let us consider the cases of her house #:

1: $3 \times 7$ (sum 10)!

2: $1 \times 2, 2 \times 6, 3 \times 4, 4 \times 8, 6 \times 7$

3: $1 \times 3, 7 \times 9$

4: $1 \times 4, 2 \times 2, 2 \times 7, 3 \times 8, 4 \times 6, 6 \times 9$

5: $1 \times 5, 3 \times 5, 5 \times 7, 5 \times 9$

6: $1 \times 6, 2 \times 3, 2 \times 8, 4 \times 9$

7: $1 \times 7, 3 \times 9$

8: $1 \times 8, 2 \times 4, 2 \times 9, 3 \times 6, 4 \times 7, 6\times 8$

9: $1 \times 9$ (sum 10)!

So, we can see that the only possible sum she could have determined was 10.

Moderator note:

Most solutions only listed out that there were multiple ways to get a certain last digit. As Maxime points out, we also have to check that they have a different sum (otherwise, we could conclude what the sum was). For example, we have $1 \times 8 = 8, 3 \times 6 = 18$ giving us the same last digit of 8, and $1 + 8 = 3 + 6$ which has the same sum.
Luckily, with the full list of possible products, we can see that the other rows have terms which give us a distinct sum.

Note that we do not know what pair of numbers Gabriel is thinking of, but we do know their sum.

"...if her house number could only be expressed as the last digit of the product of 2 numbers in exactly 1 way..." OR, if it could be expressed in multiple ways that sum to the same value.

James Tisajokt - 4 years, 1 month ago

Indeed. That was reflected in the note. Let me make it more obvious.

Calvin Lin Staff - 4 years, 1 month ago

If you try every possible combination you will, naturally, solve the problem; but I think it misses the point.

Peter Byers - 4 years, 1 month ago

@Vasco Lynch - I made the same assumption and that left me with 3 to try 3x7, 9x7 and 3x9.

Alan Parks - 4 years, 1 month ago

guys, can you tell me, in the whole problem where it says that the sum is unique and what if heather's house's number is 3? I mean in that way also a sum can be derived.

Amit Chand - 4 years, 1 month ago

If the last digit of Heather's house number was 3, Heather would not be able to know the sum as it could be 1 + 3 = 4 or 7+9 = 16 .

Siva Budaraju - 4 years, 1 month ago

0 is out since its product is always 0 and therefore her address would have to be 0. 1 is going to just be whatever number is then selected. 2 is going to be 2 4 6 8 0 repeat so 2 is out 3 is possible 4 is 4 8 2 6 0 repeat same as 2 out 5 is 0 5 repeat so out as unique 6 is 6 2 8 4 0 repeat same as 2 out 7 is possible 8 is 8 6 4 2 0 repeat same as 2 out 9 is possible

So since 1 can't be times 1 we don't have a 1 it is unique, all even numbers out still times 1, so 1 3 7 9 are possible 1 values.

Combined that is products 3 7 9 21 27 63.

The only unique products there are 9 and 21.

Both of which have a sum of 10!

Note I eliminated 2, 4, 5, 6, 8 in my head and only worked with what was left meaning much less work to the solution. Only the explanation took time.

Eric Belrose - 4 years ago

Just a quick question, the problem states: "The last digit of the product of the two numbers is your house number" So I assumed the product has to be a two digit number, is this a valid assumption? This led me to ignore the multiples of 1, and I ended up with the unique last digit 7 from 27, and therefore a sum of 12.

Vasco Lynch - 4 years, 1 month ago

You assumed incorrectly. The last digit of 2 is 2. (It is also true that the first digit of 2 is 2.)

Calvin Lin Staff - 4 years, 1 month ago

It never says that the product is a two-digit number.

Siva Budaraju - 4 years, 1 month ago

I thought that too, until I noticed 3 has the same case as 7, but they have different sum.

1
2

>>> [[i + j, str(i * j)[-1]] for i in range(1, 10) for j in range(i + 1, 10)]
[[3, '2'], [4, '3'], [5, '4'], [6, '5'], [7, '6'], [8, '7'], [9, '8'], [10, '9'], [5, '6'], [6, '8'], [7, '0'], [8, '2'], [9, '4'], [10, '6'], [11, '8'], [7, '2'], [8, '5'], [9, '8'], [10, '1'], [11, '4'], [12, '7'], [9, '0'], [10, '4'], [11, '8'], [12, '2'], [13, '6'], [11, '0'], [12, '5'], [13, '0'], [14, '5'], [13, '2'], [14, '8'], [15, '4'], [15, '6'], [16, '3'], [17, '2']]

Mehdi K. - 4 years, 1 month ago

Darmawan Putra Wijaya
Apr 24, 2017

Table of the possible 1-digit multiplication 1
2 4
3 6 9
4 8 12 16
5 10 15 20 25
6 12 18 24 30 36
7 14 21 28 35 42 49
8 16 24 32 40 48 56 64
9 18 27 36 45 54 63 72 81

Here is all the possibility, but since Gabriel say that he is thinking of two distinct single-digit numbers. It means that it cant be two same number (Ignore the number at the diagonal). Since Heather know the sum right away after Gabriel say that the last digit of the product of the two numbers is Heather's house number. The last digit must be unique. The number that have unique last digit is 9. So Heather's house number is 9, leading to to the answer 1x9.

The two number is 1 and 9, the house number is 9, and the sum is 10.

Moderator note:

Note that the conclusion of "The two number is 1 and 9" is not true. Do you see why?

Hint: The conclusion relies on the premise that "The last digit must be unique". How many unique last digits are there?

21 also has a unique last digit amongst those possibilities, and 7+3=10. She didn't say she knew the numbers, only that she knew the sum of the numbers. So the numbers are either 1 and 9, or 3 and 7, and both sets sum to 10.

Louis W - 4 years, 1 month ago

Good catch.

Joshua Brandt - 4 years ago

The last digit does not have to be unique. Two number could have the same last digit and be the product of 2 different couple that has the same sum.

maxime weill - 4 years, 1 month ago

That's a great observation to make! It happens in $8: 1 \times 8 = 8, 3 \times 6 =18$ . Thanks for highlighting this, let me make it more obvious to everyone.

Luckily, there are other cases that allow us to cancel out 8.

Calvin Lin Staff - 4 years, 1 month ago

I feel that when you say "the last digit", you are implying there is more than one number in the product. This would make 3*9 = 27 (sum=12) as the only correct answer.

Chris Ellis - 4 years, 1 month ago

I disagree. The last digit in the number 2 is 2.

Calvin Lin Staff - 4 years, 1 month ago

I came to 10. The puzzle implies that there are multiple digits in the sum, but if you eliminate 1x2, 1x3, ..., 1x9, you're left with three (3) answers: 21 (sum=10), 27 (sum=12), 63 (sum=16). So you cannot eliminate a single digit sum. I assumed that the house number is not "0", but this factor doesn't come into play. When you eliminate 1x1, 2x2, ..., 9x9 because the puzzle states "two distinct single-digit numbers", you're left with two (2) answers: 9 (sum:10) and 21 (sum: 10). Since I was partial to a multi-digit sum, I went with 7x3=21, 7+3=10.

Joshua Brandt - 4 years, 1 month ago

Charles Smart
Apr 25, 2017

slightly different tack: if we assume 0..9 are the digits, but 0 cannot be a house number, then range of distinct digits is 1..9 . if we assume the phrase 'last digit of product' implies the product is 2 digit number, then possible unique results for the last digit of product are '1' & '7' (3x7=21 and 3x9=27) . assume Heather knows her house number and so picks the right sum (if 1, 3+7=10, but if 7, 3+9=12)... but in this quiz we have still two choices left, but permitted 3 attempts to answer, so try 10 or 12 and get result required.

Why can't 0 be a house number? Also, 7 can be the last digit of $1\times 7=7$ and $3\times 9=27$ , so there is a unique answer and you don't have to check with 12.

Christopher Boo - 4 years, 1 month ago

Tell me anywhere the address is zero

Kurt Frenter - 4 years, 1 month ago

The reason zero cannot be a house number is that house numbers count up from a street intersection. Zero would mean no intersection, as the house would stop the street from intersecting. No possible counting up to zero.

Kurt Frenter - 4 years, 1 month ago

Mike Ross
Apr 26, 2017

1 is the only unique result hence the two digits are 3 and 7, sum = 10

It's not clear what your diagram implies. Can you elaborate?

Christopher Boo - 4 years, 1 month ago

Joe Pisano
Apr 25, 2017

If we are assuming the sum leads to a two-digit answer ("last digit" implies this), then 9x7 is the only pair that gives 3, so 16 is another possible answer.

No, you can't make that assumption at all because it's not stated.

Pi Han Goh - 4 years, 1 month ago

Dan Pattimore
Apr 25, 2017

7x3 is the only calculation that gives a unique final digit.

Heather must live at No. 1

Therefore the sum of these integers is 10.

9*1 also gives a unique final product She could leave at house #9 Regardless of what her house number is... 9+1 is also 10 :)

Alejandra Sierra - 4 years, 1 month ago

I found it fascinating that we could find the sum even though we don't know Heather's house number

Pranshu Gaba - 4 years, 1 month ago

You do not need to know the house number

The answer is 10.

6 solutions

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