Tyne trial

Welcome to the (virtual) Tyne Trial! Right now, various mathematicians are in Newcastle looking for clues. With a little help from Google Street View and a few photos, you can enjoy this puzzle hunt too.

There is a (virtual) locked box at mscroggs.co.uk/tyne-trial/box. Your challenge is to work out the code for the combination lock and get to the prizes inside. The code is a four-digit number; none of its digits is 0.

During the Tyne Trial, you will discover seven clues to the code. You will need to find information around Newcastle and solve some puzzles to reveal these clues. Throughout the text below, there are various links: by following these links, you will be able to find all the information you need, as well as enjoying a few of the sights of Newcastle. Each of the clues is a four-digit number, plus the "score" of the number.

The score of a number is the number of values of \(i\) such that the \(i\)th digit of the code is a factor of the \(i\)th digit of the number. For example, if the code was 1234, then the score of the number 3654 would be 3 (because 1 is a factor of 3; 2 is a factor of 6; and 4 is a factor of 4).

The Tyne Trial starts at the City Library. Head out of the library and cross the road at the lights. There is a postbox near the crossing. The post will next be collected at \(a\) o'clock on Monday.

The score of the number \(a^4\) is 2.

Head back across the road, turn right, then turn left once you reach the end of the library. Head right and under the building, then turn left.

When you reach Northumberland Street, you'll be looking at a branch of Mulysatoof that escaped from the mirror universe. If you look at the shop's name in a mirror (there's a conveniently placed advertising board), you'll see that the mirror people wrote just one of the letters backwards. Turn left (or right if you're a mirror person).

Just before you reach a junction, you'll see some guys on your left. The number of guys is \(b\).

We are going to start with a number then repeat the following process: if the number you have is a palindrome, stop; otherwise add the number to itself backwards. For example, if we start with 219, then we do: $$219\xrightarrow{+912}1131\xrightarrow{+1311}2442.$$ If you start with the number \(10b+9\), what palindrome do you get? The score of the answer is 0.

(If you start with 196, it is unknown whether you will ever get a palindrome.)

Turn right, and walk towards Grey's Monument, a column with a statue of Charles Grey at the top. The year in which the column was erected is \(c\).

There are 12 three-digit numbers whose digits are 1, 2, 3, 4, or 5 with exactly two digits that are ones. How many \(c\)-digit numbers are there whose digits are 1, 2, 3, 4, or 5 with exactly \(c-1\) digits that are ones? The score of the answer is 1.

If thinking about Earl Grey has made you want a cup of tea, pop into the kitchen and make one.

Turn left, and head down Grey Street. After crossing a road and heading straight on twice, turn right onto a street whose first letter is a Roman numeral. As you head down this street, you'll see a statue of Queen Victoria. Near the statue, there's an information board about the cathedral. The year in which two industrial pioneers both died is \(d\).

The largest value of \(n\) such that \((n!-2)/(n-2)\) is an integer is 4. What is the largest value of \(n\) such that \((n!-d)/(n-d)\) is an integer? The score of the answer is 1.

Be sure to do something to amuse Queen Victoria as you leave.

Turn left and head down St Nicholas Street towards Newcastle Castle. Walk past The Black Gate and under the bridge. Newcastle Keep will be on your left. The year in which (most of) the castle was leased to Alexander Stephenson is \(e\).

We are going to start with a number then repeat the following process: if we've reached 0, stop; otherwise subtract the smallest prime factor of the current number. For example, if we start with 9, then we do: $$9\xrightarrow{-3}6\xrightarrow{-2}4\xrightarrow{-2}2\xrightarrow{-2}0.$$ It took 4 steps to get to 0. What is the smallest starting number such that this process will take \(e\) steps? The score of the answer is 1.

Keep heading straight on, onto the bridge under the train line. You can see many of the other bridges of Newcastle, including the famous Tyne bridge. If you're lucky, a train might pass over the bridge above you. If you're very lucky, you might see the swing bridge to your left open to let a boat pass through.

Turn around and head back past the Keep, and under the bridge to the Black Gate. Turn left and head down Westgate road. When you reach a junction, head left to the statue of George Stephenson. Near the statue, there is a bar called Liberty House (formerly The Union Rooms). The number above Liberty House's door is \(f\).

I thought of a four digit number. I removed a digit to make a three digit number, then added my two numbers together. The result is \(200f+127\). What was my original number? The score of the answer is 1.

Head back away from Stephenson's monument, turn left and head down Pudding Chare. At the end of Pudding Chare, turn left along Bigg Market, then right along Grainger Street.

Near the corner of Grainger Street and Nun Street, there's a plaque in the pavement. Follow Mrs Mary Howard's example, and adjust your hat in the window. If you didn't think you were wearing a hat, then your invisible hat is probably wonky and in need of adjustment.

On the floor at the entrance to Grainger Market, you might spot this circle thorem:

Turn left and head down a street that starts with the same Roman numeral as earlier. The value of this Roman numeral is \(g\).

If you roll two six-sided fair dice, the most likely total is 7. What is the most likely total if you rolled \(1470+g\) dice? The score of the answer is 3.

You should now have enough information to work out the code. Head to mscroggs.co.uk/tyne-trial/box.