# Beginner puzzle

## This puzzle is intended to be suitable for people who are new to puzzle solving.

### Clarification: Both experienced solvers and new solvers are welcome to post solutions to this puzzle.

Fill each square in this cross-number with a non-zero digit such that all of the conditions in the clues are satisfied. The digits used are not necessarily distinct.

1 ACROSS. A prime which is the sum of two squares
3 ACROSS. Twice the answer to 2 DOWN

1 DOWN. $$p \times q$$, where $$p$$, $$q$$ are prime and $$q=p+4$$
4 DOWN. $$60$$% of 5 ACROSS

This puzzle comes from the Senior Kangaroo 2021 contest.

A similar puzzle is here: A simple cross-number puzzle

1 Down:

$$p$$ cannot exceed 10 since $$pq$$ would be too large otherwise (not 2 digits). The possibilities are $$p = 3, q = 7, pq = 21$$ and $$p = 7, q = 11, pq = 77$$.

From 1 Down, 3 Across starts with either 1 or 7. Since 3 Across is twice 2 Down and the two are equal in length, 3 Across cannot start with 1, and 1 Down is 77.

7 ?
7 ? ?
? ?

1 Across:

By Fermat's two-square theorem, a prime that is a sum of two squares is precisely a prime that is 1 modulo 4. The only 2-digit prime that is 1 modulo 4 and starts with 7 is 73 (= 64 + 9).

7 3
7 ? ?
? ?

Alternatively, you can

deduce the top middle cell by seeing that half of 3 Across (7xx) must start with 3. This way you can simply ignore the clue for 1 Across.

4 Down & 5 Across:

4 Down is 3/5 times 5 Across, and the two share the last digit. 5 Across must be a multiple of 5 (so that 3/5 of that is an integer), and then so is 4 Down (since the last digit is either 0 or 5). Therefore, 5 Across is a multiple of 25, i.e. one of 25, 50, or 75. 4 Down can be one of 15, 30, or 45.

Since 3 Across is twice 2 Down, it is an even number, and it ends with an even digit. So 4 Down must start with an even digit, making it 45.

7 3
7 ? 4
7 5

3 Across & 2 Down:

We can write down an equation with a single unknown being the missing digit: $$2(307 + 10x) = 704 + 10x$$. Solving this gives $$x = 9$$.

Solution:

7 3
7 9 4
7 5

• Well explained! Commented May 13 at 5:12

A bit of different approach(more straightforward/bruteforced, less math knowledge) than Bubbler's answer:

1 ACROSS. A prime which is the sum of two squares

1 across is 2 digits, so we can have values between 13($$3^2 + 2^2$$) and 98($$7^2 + 7^2)$$

3 ACROSS. Twice the answer to 2 DOWN

2 down must be at most 499(otherwise 3 across would become 4 digits). That means that 1 across can only end in 1, 2, 3 or 4
Using Excel and conditional formatting, I limited the possible values for 1 across to the following:
13, 32, 34, 41, 52, 53, 61, 64, 72, 73, 74, 81, 82

1 DOWN. p×q, where p, q are prime and q=p+4

First few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23
With difference of 4 we get a few groups: (3, 7), (7, 11), (13, 17), (17, 23)
We also see that $$13*17=221$$, but 1 down is only 2 digits, so we only have 2 options for 1 down: 21 or 77
But given that 1 across can not start with 2, the only possible answer for 1 down is 77.
That also limits 1 across to 72, 73 or 74
Also, hint 2 suggests that 3 across(that starts with 7) is double of 2 down. That leaves only 1 possibility of 1 across = 73(because highest double of 2xx is $$299*2 = 598$$ and lowest double of 4xx is $$400*2 = 800$$)

4 DOWN. 60% of 5 ACROSS

With both numbers being 2 digits and sharing the last digit, we can limit the possible values to:
(25, 15), (75, 45)
So first digit of 4 down is either 1 or 4, but 3 across is double of 2 down, so the last digit of 3 across must be even. That leaves us with only 1 possibility for 4 down: 45
That means that 5 across is 75

So far we have:

7 3
7 x 4
 7 5

Now to find the remaining digits:

3 x 7 +
3 x 7
---
7 x 4

 14
 2x
6

Last digit of 2x+1 = x and 2x+1 must be between 11 and 19(otherwise the sum would be either 6x4 or 8x4)
Therefore x = 9(18+1 = 19)

7 3
7 9 4
 7 5

And fit it back to the hints:

1 ACROSS. A prime which is the sum of two squares

$$8^2 + 3^2 = 64 + 9 = 73$$

3 ACROSS. Twice the answer to 2 DOWN

$$397 * 2 = 794$$

1 DOWN. p×q, where p, q are prime and q=p+4

$$p = 7, q = 11, p*q = 7 * 11 = 77$$

4 DOWN. 60% of 5 ACROSS

$$75 * 60% = 45$$