### Dozens

Can you select the missing digit(s) to find the largest multiple?

### Clock Squares

Can you find a way of predicting the value of large square numbers with the help of our power modulo calculator?

### Last Biscuit

Can you find a strategy that ensures you get to take the last biscuit in this game?

# Clock Arithmetic

##### Age 11 to 18Challenge Level

Thank you for all the fantastic solutions we received to this problem. Since the answers to tasks 1-5 were shown in the problem, here we have only included the solutions to task 6.

If today is a Monday, what day will it be in $31$ days' time?

Ilham from St. Aidan's Catholic Primary Academy in England, Jerry, Lucas and Jaa from Rugby School Thailand and Michelle from Westridge School in the USA used the language of days and weeks. Ilham wrote:

31 days contain 4 weeks (28 days) and 3 days and 3 days onward from Monday is Thursday so in 31 days' time, it will be Thursday.

Aaron from Harrow International School Hong Kong used remainders:

Days in a week: 7
Remainer of (31 days $\div$ 7 days of the week) = 3
7 $\times$ 4 is 28
31 $-$ 28 is 3
Monday plus 3 days is Thursday

Nayanika fom The Tiffin Girls' School in England and Lawson from Aiglon College in Switzerland used a diagram. This is Nayanika's diagram:

Sunhari and Miraya from Heckmondwike Grammar School in the UK used the same idea and  the language of modular arithmetic. This is Sunhari's work:

Monday is considered as 0 on the modulus 7 clock, and we start counting
clockwise from there.
No. of Days = 31
31 is congruent to 3 mod 7 [7$n$+$a$ where $n$=4 and $a$=3(by dividing 31 by 7 and finding the remainder)]
Therefore, the answer is (Monday+3) Thursday.

Airi from Japan, Zac from Aiglon College, Emily from The Archer Academy in England, Martin from Prague British International School Prague Libuš in the Czech Republic, Olivia from Roedean in the UK, Mahdi from Mahatma Gandhi International School in India and Professor Pat, Professor Andrew and Peter from Rugby School Thailand also used modular arithmetic to get the correct answer.

If April $1^{\text{st}}$ is a Thursday, what day will Christmas day be on?

Ilham used weeks and days language:

30 days + 31 + 30 + 31 + 31+ 30 + 31 + 30 + 24 equals 268 days
268 days $\div$ 7 = 38 weeks and 2 days and 2 days onwards from Thursday is... Saturday! (our answer).

Lawson used a diagram:

Miraya, Airi, Nayanika, Zac, Emily, Olivia, Mahdi, Professor Pat and Professor Andrew, Peter and Jerry, Lucas and Jaa  got the same answer using modular arithmetic. Airi wrote:

Counting from April 1st, there are 29 days in April, 31 days in May, 30 days in June, 31 days in July, 31 days in August, 30 days in September, 31 days in October, 30 days in November, and 25 days in December until Christmas day. (Remember not to overcount! This goes for Problem 3 and 4 as well.)
$29+31+30+31+31+30+31+30+25$
$= 309+((-1)+1+0+1+1+0+1+0+(-5))=270-2=268$
$268\equiv2\text{ (mod 7)}$
The day that is two days from a Thursday is a Saturday.

The addition might be easier if you use modulo 7 earlier:
$29+31+30+31+31+30+31+30+25$
$\equiv 1+3+2+3+3+2+3+2+4\text{ (mod 7)} \equiv 2 \text{ (mod 7)}$

If Jan $1^{\text{st}}$ is a Friday, and it is a leap year, what day will Jan $1^{\text{st}}$ be on next year?

Sunhari, Airi, Nayanika, Zac, Ilham, Emily, Martin, Olivia, Mahdi, Professor Pat and Professor Andrew and Peter answered this correctly using modular arithmetic. This is Olivia's work:

Interestingly, we can also know when 1st Jan will be a Friday again.

Exactly 1 year later, we get Sunday. Now, this would be non-leap year with 365 days ≡ 1 mod 7. Hence, 2 years later we get to Monday. This would repeat again for two more non-leap years. And so, 4 years later, 1st Jan is Wednesday. Now the fifth year would be 366 days ≡ 2 mod 7, and moving two days from Wednesday gives Friday again! So we get 1st Jan as a Friday again after 5 years.

Can you use today's date to work out what day of the week your birthday will be on?

Airi: Say that my birthday is the 12th of January. Today’s date is Friday, 26 March 2021. Counting from today, there are 5 days left in March, 30 days in April, 31 days in May, 30 days in June, 31 days in July, 31 days in August, 30 days in September, 31 days in October, 30 days in November, 31 days in December, and 12 days in January until my birthday.
5+30+31+30+31+31+30+31+30+31+12=5+309+(0+1+0+1+1+0+1+0+1)+12
=5+270+5+12=292
292≡5 (mod 7)
The day that is five days from a Friday is a Wednesday.

Sunhari: Today's date = March 25, 2021
Birthday = January 30, 2022
No of days= 6 + 30 + 31 + 30 +31 + 31 + 30+ 31 + 30 +31+30
=311
311 is congruent to 3 mod 7
My birthday will fall on (Thursday+3) Sunday.

Aaron: Today's date is Tuesday 30th of March while my birthday is on the 28th of September which is 182 days difference. 182 $\div$ 7 (since 7 days in a week) = remainder 0. Tuesday + 0 days = Tuesday
Since, April 30 days, May 31 days, June 30 days, July 31 days, August 31
days and September 28 days.
1 + 30 + 31 + 30 + 31 +31 + 28
1 +    61      +      61    +    59 = 182
28th September will also be a Tuesday

Miraya: Today is 1st April and my birthday on 20th August. So that would be 141 days. Which is 20 weeks and 1 day. That would be 1 mod 7. So 20th August would be FRIDAY.

Nayanika:

Lawson:

Zac:

Ilham: I don't know about YOU but my birthday is on the 21st of July and today is the 28th of April so, using the same method, we will conclude that this year my birthday will be on a  Wednesday.

Emily: I worked out that because my birthday is on March 1st, it is 305 days away from April 30th.
Then, I found the multiple of 7 that is 305 or the closest one below. This is 301, which is 4 below 305, meaning 305 ≡ 4 mod 7.
I had already assigned the number 4 to Tuesday, so if April 30th is a Friday, March 1st will be on a Tuesday.

Martin: Today is a Tuesday, It is the 11th of May 2021. (next year Is not a Leap year) My birthday is on the 4th of March. My birthday is in 298 days.
If we use the mod7 pattern, we notice that 298 ≡ 4 mod7 (42$\times$7+4). This means that if today is a Tuesday, my Birthday would be a Saturday.

Michelle: My birthday is on December 1. Like I solved the other problems, I found the difference between today’s date and my birthday.
Next, I used that to find 216 (amount of days until my birthday) mod 7. The remainder [of 6] was added on to today's day, Thursday, May 6th, and I found out that my birthday is on a Wednesday.

Olivia:

Mahdi: Today is 13 May, Thursday. My birthday is on 1st October.Each month has these days:
May - 18
June - 30
July - 31
August - 31
September - 30
October - 1

141 days ≡ 1 mod 7, 1 day from Thursday is Friday. And 1st October 2021 is a Friday indeed.

Professor Pat and Professor Andrew:
Professor Pat’s BirthDay:
Today is friday 21/5/2021
My Birthday is on 25/1/2022
Tuesday    249 ≡ 4 mod 7        Friday-Saturday-Sunday-Monday-Tuesday

Professor Andrew’s BirthDay:
Today is friday 21/5/2021
My Birthday is on 28/1/2022
Friday        252 ≡ 0 mod 7        Friday

Jerry, Lucas and Jaa (including another method based on the idea that 365 ≡ 1 mod 7):
Jerry=Saturday
I have worked this out by the solutions to this: today is Friday and it’s 21/5/2021, my birthday is on the date of 04/12/2021, today to 12/04 is 10+30+31+31+30+31+30+4=197, it’s divided by 7 and we get 28, left with
1, the next day for Friday is Saturday,s o that’s the working.

Jaa=Tuesday
My birthday this year was on a Monday so it shifts one day.
Monday-Tuesday.

Another way I worked it out is, finding the amount of days from today to my birthday. Divided the number by seven, and then counted from Friday to Tuesday because the remainder of the amount of days divided by 7 is 4. So just like the mod 7 clock. You shift 4 spaces clockwise.

Lucas=Monday

Last time my birthday is Sunday, this year is not a leap year so we add one.