In the latter case, you will not have leaked any information to Annabel, nor will she have leaked anything to you. But today is different since I am asking you to contort your brain in a way you have probably never contorted it before. And there are many solutions, each with strengths and weaknesses. STEP 2 You write down the number of the person you suspect on a piece of paper. Annabel does the same. STEP 3 You hand Dan the two pieces of paper, and you ask him to tell you if both pieces have the same number.
If Dan says the numbers are the same, you know the suspects are the same. If the numbers are different, the suspects are different, and neither you nor Annabel are any the wiser about who the other person suspects. Job done! First, it relies on there being a Dan, i. More importantly, you are both revealing some information to Dan. The whole point of this exercise is to never reveal anything to anyone. Is there a way that does not involve anyone else?
Feel free to make suggestions below the line about how you might want to approach the problem. I mentioned at the top that the Abel Prize honoured one of the pioneers of the zero-knowledge proof. In the s, Wigderson, together with Oded Goldreich and Silvio Micali, showed that any true mathematical statement has a zero-knowledge proof.
At the time, this result was a purely theoretical advance, but in recent years zero-knowledge proofs have found important applications in digital security, especially in cryptocurrencies and authentication systems. The zero-knowledge proof allows two people to establish trust while revealing nothing. If you are interested in finding out more about the work honoured by the Abel prize, you may enjoy this short video I made for the Abel Prize that was broadcast during the announcement ceremony.
I set a puzzle here every two weeks on a Monday. If you would like to suggest one, email me. Bitcoin solves this problem with the following idea. As the total computing power increases from 10 computers and a billion computers in the above example , we will increase the hardness of solving the puzzle.
So how exactly is the puzzle hardness varied? Remember that in a target hash value, only some first x number of bits need to be zeroes and the other bits are unspecified? We can vary the hardness of solving the puzzle by varying x. Increasing x increases hardness. Pretty much half of all the hash values have the first bit as a zero and the other half have the first bit as one. Solving the puzzle in this case can be quite easy, since every other input will succeed as a solution.
On the other end of the spectrum, if all the bits are zeroes, then the puzzle becomes the same as solving the one-wayness of the hash function, which we know is extremely hard. So you can observe that spectrum of hardness; the knob here is the value of x. So, as more people join the Bitcoin network, we can ensure that there is usually only one block producer by increasing the puzzle hardness.
The total computing power in the network is called the hash power or the hash rate of the network. Hash Functions in Bitcoin. Energy Consumption, Block Rewards,. Mining, Mining Pools, Good-to-know Facts. The total computing power on the network trying to solve these puzzles is called the hash power or the hash rate. Blockchain Concepts. Hash Value as Pointers.
Imagine if you invented a digital currency yourself called the green blob currency and sent your friend 10 digital green blobs. You would have a big problem — your friend could simply cut and paste your green blobs and send everyone in your school 10 green blobs each. This is a fundamental problem which needs to be solved before you can create a virtual digital currency. This is a register of all transactions to date, which everyone can see.
The blockchain shows when Bitcoins are created, and when they pass between individuals. Bitcoin have a very clever way of making sure that the blockchain is accurate. The blockchain is checked for accuracy by computers that solve huge number crunching problems. Bitcoin relies on thousands of computers across the world solving complicated problems which verify the transactions and prove that the blockchain is accurate.
The calculations involved require very powerful computers which cost a lot of money, and they also require storage space and electricity to operate. Bitcoin needs to provide an incentive for people to solve these number crunching problems and it does this by rewarding people with new Bitcoins when they solve a problem. This is the only way that new Bitcoins are created. The value of a Bitcoin is simply determined by the laws of supply and demand — there are a limited number of Bitcoins in circulation, and therefore their price is decided by how popular they are, and how many people want to trade in Bitcoins at that time.
The chart below shows how the price of Bitcoins changed over a two year period. The price varied so much that it will certainly make a lot of investors wary about losing their money! One really interesting thing to note about Bitcoins is that they are designed so that over time fewer new coins will be mined.
Around the year there will be no more new Bitcoins created. From an economics point of view this creates some very interesting questions about what will happen to the value of Bitcoins in the future. Bitcoin can only function because of the clever mathematics which is in the background enabling it to exist.
Since its creation in Bitcoin has created intense media coverage and speculation as to what will happen with this new currency. Will Bitcoin be the currency of the future, will it make people rich, or collapse overnight? Or will Bitcoin remain as a currency which only a minority of people are willing to invest and trade with? Watch this space! While only a tiny minority of people actually make their living from Bitcoin at the moment, there are lots of technical jobs available which will use similar skills.
If you are fascinated by how virtual currencies work then you might be interested in a career as an economist, where you can examine the role of economic factors such as supply and demand as part of your work. If you want to study economics at university you will need to study A-level Mathematics. You might also like to consider studying Further Mathematics A-level.
Equally you might be interested in the high end computing which is behind Bitcoin, or finding out more about the complex mathematics involved. How many ways can you create a code for the vowels by assigning to each vowel a different vowel? Write down all the possible questions that could have been asked if this was the diagram provided in a mathematics textbook. I really enjoy the puzzles.
The flips when stopped were two Heads and one Tail. I get the same answer but intermediate steps differ. Am I looking at this incorrectly? The only reason I chose to list the next two outcomes was to produce equally likely outcomes making the arithmetic very slightly easier. I am glad you enjoy the puzzles. Six Yellow are working together using Transum Maths this morning Great second lesson with our Shanghai pupils and Y6.
Transum problems making their brains hurt! Great to see the resilience and enjoyment in the problems. RanbyHouse pic. Thank you. Do you have the answers? Yes, each of the puzzles has an answer that appears when you are logged in as a subscriber. You can find a subscription application form here. You offer great exercises, puzzles, review material, and so much more. I really enjoyed using Transum Math. Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world.
Click here to enter your comments. Code Cracker Crack the code by replacing the encrypted letters in the given text. Prime Numbers Jigsaw Interactive jigsaw puzzles of different types of grids containing prime numbers. Thai Numbers Jigsaw An online interactive jigsaw puzzle of a grid of Thai number symbols.
Chinese Numbers Jigsaw An online interactive jigsaw puzzle of a grid of Chinese number symbols. How Many Squares? Vocabero Find the mathematical word from a series of guesses and clues. Equatero Find the expression from a series of guesses and clues. Fleur-De-Lis Click on six fleur-de-lis to leave an even number in each row and column. Tower of Hanoi Move the pieces of the tower from one place to another in the minimum number of moves.
Ludicross Arrange the given numbers on the cross so that the sum of the numbers in both diagonals is the same. Largest Product A drag and drop activity challenging you to arrange the digits to produce the largest possible product. Congruent Parts Use the colours to dissect the outlines into congruent parts. Similar Parts Use the colours to dissect the outlines into similar parts. Overloaded Fraction A set of ten puzzles requiring you to arrange the given digits to make an equivalent fraction.
Product Square Arrange the given numbers in a three by three grid to obtain the diagonal, row and column products. Clue Sudoku A different way to complete a Sudoku puzzle with clues available at every stage. Plane Numbers Arrange numbers on the plane shaped grid to produce the given totals Cryptographic Fill in the squares according to the clues given by the string of numbers for each row and column.
Unbeknownst Some picture grid puzzles which can be solved by using simultaneous equations. Partial Pyramids Calculate the missing numbers in these partly completed pyramid puzzles. Pyramid Puzzle Numbers in the bricks are found by adding the two bricks immediately below together.
The Broken Chessboard Puzzle The chessboard has been broken into 13 pieces. Can you put it back together? Doughnut Dissection A puzzle to find four different ways of making by multiplying together three different numbers. Power Shift Arrange the given numbers as bases and indices in the three-term sum to make the target total. Satisfaction This is quite a challenging number grouping puzzle requiring a knowledge of prime, square and triangular numbers.
Squorder The Transum version of the traditional sliding tile puzzle. Mine Find Find where the mines are hidden without stepping on one. Pentominoes Arrange the twelve pentominoes in the outline of a rectangle. Number Jigsaws Online, interactive jigsaw puzzles of grids of numbers. Roman Numerals Jigsaw An online interactive jigsaw puzzle of a grid of Roman numerals.
Magic Square Jigsaws Interactive jigsaw puzzles of four by four magic squares. Plus A number arranging puzzle with seven levels of challenge. Area Maze Use your knowledge of rectangle areas to calculate the missing measurement of these composite diagrams. Car Park Puzzle Can you get your car out of the very crowded car park by moving other cars forwards or backwards? One Digit Only Find expressions using only one digit which equal the given targets.
Centexpression Arrange the numbers from 1 to 9 to make an expression with a value of Perfect Magic Square Arrange the sixteen numbers on the four by four grid so that groups of four numbers in a pattern add up to the same total. Octagram Star Arrange the sixteen numbers on the octagram so that the numbers in each line add up to the same total. Hexagram Star Arrange the twelve numbers on the hexagram so that the numbers in each line add up to the same total.
Triangled Hexagram Arrange the twelve numbers in the triangles on the hexagram so that the numbers in each line of five triangles add up to the same total. Awe-Sum Arrange the given digits to make six 3-digit numbers that combine in an awesome way. Goal Products Arrange the numbered footballs on the goal posts to make three, 3-number products that are all the same.
Circumfraction Quite a challenging number placing puzzle involving fractions. Magic Square Each row, column and diagonal should produce the same sum. Unmagic Square Like the magic square but all of the totals should be different.
One Torch Tunnel Solve the problem of getting four people through a tunnel with one torch in the minimum amount of time. Separated Twins Can you find a 6 digit number containing two each of the digits one to three which obeys the rules given? Without Lifting The Pencil Can you draw these diagrams without lifting your pencil from the paper? River Crossing The traditional River Crossing challenge.
Can you do it in the smallest number of moves? Digivide Arrange the numbers from 1 to 6 in the spaces to make the division calculation correct. Scouts in Boats Arrange a rota for the Scouts to travel in boats so that they are with different people each day. Quad Areas Calculate the areas of all the possible quadrilaterals that can be constructed by joining together dots on this grid.
Four Sum Arrange the given number tiles to make two 2 digit numbers that add up to the given total. Path Puzzle A great puzzle requiring you to use all of the cards to create a continuous red line from start to finish. Lemon Law Change the numbers on the apples so that the number on the lemon is the given total. Scheduling Puzzle Make a schedule for the hour Darts Marathon which will take into account everyone's requests and keep everyone happy.
Online Psychic Let the psychic read the cards and magically reveal the number you have secretly chosen. Numskull Interactive, randomly-generated, number-based logic puzzle designed to develop numeracy skills. Addle Arrange the numbers from 1 to 14 in the spaces to make the sums correct.
How fast can you do it? Suko Sujiko Interactive number-based logic puzzles similar to those featuring in daily newspapers. Latin Square Puzzles Arrange the given digits to make a Latin square with the given row and column calculation results. Polygon Pieces Arrange the nine pieces of the puzzle on the grid to make different polygons. Pu Wiang A fewest-moves, counter-swapping challenge invented in northern Thailand. Frustration A logic challenge requiring a strategy to update each of the numbers in a grid.
Prime Square Drag the numbers into the red cells so that the sum of the three numbers in each row and each column is a prime number. Mixpressions Arrange the cards to create a valid mathematical statement. Pancake Day Toss the pancakes until they are neatly stacked in order of size. Word Search Find the mathematical words in the grid of letters.
Vector Cops Help the cops catch the robbers by finding the vectors that will end the chase. Triside Totals Arrange the digits 1 to 9 on the triangle so that the sum of the numbers along each side is equal to the given total. Multitude Arrange the given digits to make three numbers such that the third is the product of the first and the second. Double Treble Arrange the digits to make three 3 digit numbers such that the second is double the first and the third is three times the first.
Not Too Close The students numbered 1 to 8 should sit on the chairs so that no two consecutively numbered students sit next to each other. Tools In how many different ways can the numbers be arranged to give the same totals? Window View Drag the 20 flowers into the gardens so that 9 flowers are visible from each window of the house. Shapes In The Stars Join up the stars to find the hidden regular polygons.
Satisfy Place the nine numbers in the table so they obey the row and column headings about the properties of the numbers. T Puzzle Use the pieces of the T puzzle to fit into the outlines provided. Shunting Puzzles Move the trams to their indicated parking places in the shunting yard as quickly as possible.
Magic Square Puzzle Find all of the possible ways of making the magic total from the numbers in this four by four magic square. Cubical Net Challenge Find all the ways of painting the faces of cubes using only two colours. Nine Digits Arrange the given digits to make three numbers such that two of them add up to the third.
Tangram Table Use the pieces of the tangram puzzle to make the basic shapes then complete the table showing which shapes are possible. Calculator Words Turn your calculator upside down to make words out of the answers to these questions. Vowelless Vowels have been taken out of mathematical words. Can you recognise them? Pick Up Sticks If you were to pick up the sticks from this pile so that you were always removing the top stick what calculation would you create?
Divisive Arrange the digits one to nine on the spaces provided to make two division calculations containing multiples of three. Go Figure Arrange the digits one to nine to make the four calculations correct. Olympic Rings Place the digits one to nine in each of the regions created by the Olympic rings so that the sum of the numbers in each ring is the same. Arithmagons Find the missing numbers in these triangular, self-checking puzzles and discover the wonders of these fascinating structures.
Area Wall Puzzles Divide the grid into rectangular pieces so that the area of each piece is the same as the number it contains. Xmas Ornaments A hands on activity requiring students to arrange Christmas ornaments in a square box. Pentadd Quiz Find the five numbers which when added or multiplied together in pairs to produce the given sums or products. Prime Labyrinth Find the path to the centre of the labyrinth by moving along the prime numbers. Truculent Can you arrange the seven counters on the grid despite their truculent behaviour?
Maths Crossword An interactive mathematical crossword for you to do online. Tantrum A game, a puzzle and a challenge involving counters being placed at the corners of a square on a grid. Scallywags and Scoundrels Arrange the scallywags and scoundrels on the chairs so that the numbers of any two sitting next to each other add up to a prime number.
No Partner Find which numbers in a given list do not combine with other numbers on the list to make a given sum. Thrice Can you arrange all of the counters on the grid to form 10 lines of three counters? Dominoes Puzzle Arrange the dominoes in seven squares. The number of dots along each side of the square must be equal to the number in the middle The Miller's Puzzle This is an interactive version of the puzzle described by Henry Ernest Dudeney in The Canterbury Puzzles Delightfully Divisible Arrange the digits one to nine to make a number which is divisible in the way described.
Nine Nine Nine Use the digits 1 to 9 to make three 3 digit numbers which add up to Spinsum Arrange the numbers on the squares so that the totals along each line of three squares are equal. Pattern Clues An interactive activity challenging you to reproduce a pattern of coloured squares according to given clues. Brainbox A puzzle requiring the arrangement of numbers on the function machines to link the given input numbers to the correct output. Zygo Interactive, randomly-generated, number-based logic puzzle designed to develop numeracy skills.
Online Sudoku An online, interactive version of the popular number placing puzzle. Puzzle Cube Net A jumbled moving-block puzzle cube is shown as a net. Can you solve it? Where's Wallaby? Find the hidden wallaby using the clues revealed at the chosen coordinates. Jugs Can you make 4 litres if you only have 7 and 5 litre jugs? Sisters Outnumbering Brothers Work out the total number of children in Rachel's family using the clues about brothers and sisters. Time and Calendars How many times in a 24 hour period do all four digits on my bedside clock change at the same time?
Santa's Small Window How do you increase the area of a window without changing its height or width? How Much Richer? A puzzle about the relative wealth of Maureen compared to Doreen. Factor Shopping Find the prices of the items bought if they are different and all factors of the total bill.
Splitting Ten for a Product Which of the many ways of splitting ten gives the largest product? Where's the Father What seems like a familiar puzzle has an unusual twist. A Frobenius coin problem What is the largest amount that cannot be made with 5p and 7p coins? Chris and his Unpredictable Kids A think of a number puzzle with an unpredictable element. Keith and Kath Find the ages of Keith and Kath from the clues given.
Two-Faced and Blue How many of the smaller cubes that make up the 3x3 cube have two faces painted blue? Santa is Cool What temperature is reported using the same number in Celsius and Fahrenheit? Three Trios of Triplets Thrice A simple question that seventy two percent of a group of people got wrong.
Trains Together How far apart will the trains be half an hour before they meet? Six Ropes Can the pirates win their freedom by tying the six pieces of rope into one large loop? What's My Name Riddle Work out the name from the rhyming riddles. Eva's Eggs and Fickle Fractions How many eggs did Eva take to market in order to make those strange half sales? Feeding Fools and Horses How long will the horse feed last the horses after some of them left the stables?
Balancing Balloons How many balloons must Jamie give to Ben to balance the balloon equation? A Puzzle from Carl Carl's puzzle is about the result of dividing the product of his parents' ages by double his own age. Odds from evens up to Subract the sum of the odd numbers less than from the sum of the even numbers less than Noel in Lapland Work out the length of Noel's trip to Lapland given the details of the weather conditions.
Multiple Remainders What is the second smallest number such that when it is divided by 5 the remainder is 4, and when divided by 7 the remainder is 6? Percy Cod's Kids Percy Cod was talking about his children. Carrot-eating Critters Work out how many of each kind of animal are in the field from their carrot-eating habits. Red Arrow Square What fraction of the square does the red arrow cover?
Monkeys, Kittens and Dogs Who is most likely to be able to work out the square root of ? Feed The Horses How long will the remaining feed last the horses that have not been sold? Prime Permutations Of all the permutations of 1 to 9 used to make nine-digit numbers, how many are prime? Choir Eye Colour Figure out the percentage of choir members that do not have blue eyes from the clues given. Calculator Keys at the Corners of a Rectangle A question about the four keys at the corners of a rectangle on a calculator.
Average House Numbers Work out the house numbers from the clues about the mean, median and mode. Clock in the Mirror What time was the clock in the mirror really showing? Delightfully Divisible Find a pandigital number that is delightfully divisible. Five integers with a product of 12 Can you find five integers that multiply together to give twelve? Square in a Rectangle What is the largest square that can be drawn in the corner of a 10cm by 15cm rectangle?
London Marathon A question about the average speed required for the second half of the marathon. Cutting the Lawn Aynuk and Ayli cut half of the lawn each. The Missing Pound's Found A really wonderful answer to the missing pound puzzle. Forty Five in Four Parts Split the number 45 into four parts according to the given information. Shrivelled Spuds Work out the weight of the potatoes after they have been left out in the sun to dry.
Transposition Error Work out the bank balance given information about the transposition error. Counting Sheep Work out the number of sheep owned by Percy and Patsy from the given clues. Juggling with Egg Timers Can you time exactly nine minutes using the four and seven minute egg timers? Permutation Sum What is sum of all the four digit numbers containing all of the digits one to four?
Odd Probability What is the probability that two random numbers are both even if they are not both odd? Regions in Circles Calculate the number of regions in a circle formed by intersecting chords. Brothers and Sisters Can you work out the number of children in the Numlove family given the clues about brothers and sisters? Cake Cut Where should the last slice of cake be cut to give two equal pieces?
Holy Sphere Calculate the remaining volume of the sphere after a cylindrical hole has been drilled through the centre Newsletter Podcast. Jumping Flea How many different places could the flea find itself after 8 foot-long jumps either north, south, east or west? Last Digit How many positive two-digit numbers are there whose square and cube both end in the same digit?
Central Station The probability that the next train to leave will be going north is five times the probability that the next train to leave will be going south. Canteen Queue Is it possible to answer the question if Betsy's age is not known? Separated Twins Work out the combination of the safe given the clues about pairs of numbers. Divisible By Three A puzzle about two digit numbers that can be made from ten different digits.
Letters In Numbers A brand new puzzle involving the letters in numbers when written as words. Square Angled Triangle The angles of a triangle are all square numbers. Tri-Junction Puzzle What is the probability of the three cars arriving at the road junction not being involved in an accident? Two Prime Squares What is the smallest square number greater than one that cannot be expressed as the sum of two prime numbers?
The Missing Pound Where did the missing pound go in this story about three people visiting a restaurant? Ticks, Tocks, Tacks and Tucks Find how ticks compare to tocks, tacks and tucks from the given information. The Power of Christmas A question about indices to get you thinking mathematically at this festive time of year.
Ant and Dec What single question could Dec ask Ant to find out what he is thinking? Bookworm How far does the bookworm travel on its meal-time journey? Three Mathematicians How can the third mathematician be so certain that everyone wants a drink? Unfinished Game If the coin tossing game was cut short how would you share the winnings? Alphanumbetical Questions about the letters used to make the natural numbers. Best Dice Which of the unusual dice would you choose to give you the best chance of winning the prize?