Enigma 1152: Tet on the Nile

From New Scientist #2308, 15th September 2001 [link]

Pharaoah Tetrakamun was bored with the rectangular [based] pyramids at Gizah so he commanded his architect to design him a tetrahedral monument. The six edges (all different) were to measure cubes of successive integers, in cubits. The face with the largest perimeter was to form the base. Eventually the architect found the lowest possible set of six successive cubes.

In ascending order, what were the lengths of the three sides of the base?

[enigma1152]

Enigma 1185: Interval music

From New Scientist #2341, 4th May 2002 [link]

The relative frequencies of the notes of the diatonic music scale are:

C = 24, D = 27, E = 30, F = 32, G = 36, A = 40, B = 45, C’ = 48.

I recently built a series of oscillators so that I could hear how it sounded, but something went wrong with the wiring, so that I ended with an eight-note keyboard where only two were in the correct places.

However, when I played each key in order from the left, I noticed that the only intervals between adjacent notes were fourths (frequency ratio 4/3 or 3/4), fifths (3/2 or 2/3), or sixths (5/3 or 3/5).

Further, the interval (frequency ratio) between the two notes I did get right was one of these.

(a) If I played the left-hand key, what note letter did it sound?

(b) Could the notes have been arranged according to the same rules and have none in the right place?

(c) Could the notes have been arranged according to the same rules and have had more than two in the right place?

[enigma1185]

Enigma 1232: Precocious segment

From New Scientist #2388, 29th March 2003 [link]

The digits on my calculator are made up of seven segments which light up in standard ways to form numbers.

For example, to form an 8 all seven segments light, but to form a 3 only five do. I have set the calculator to calculate to five figures after the decimal point, in which case it displays trailing zeros as well as a leading zero (for example it shows 1/25 at 0.04000) and rounds the last digit in the conventional manner. I have used it to calculate the reciprocals of prime numbers. All were correct as far as 1/13, but I immediately saw that the result for 1/17 was wrong although it looked like a valid number, and I realised that one of the 42 segments was lighting which should not.

If I continue to calculate the reciprocals of the next one thousand primes, how many more answers will be wrong?

[enigma1232]

Enigma 1260: Latin fives

From New Scientist #2416, 11th October 2003 [link]

Your task today is to put one letter C, L, X, V or I into each of the 25 squares in the grid so that each row and each column (read downwards) forms a different valid Roman numeral. The sum of the numbers in the rows is to equal the sum of the numbers in the columns. Use the smallest number of Cs you can. As a hint, one of your diagonals will be a valid Roman numeral and the other will contain only one letter.

What are:

(a) the sum of the numbers in the rows (or columns); and
(b) the Arabic value of the valid diagonal?

[enigma1260]

Enigma 1261: It’s all Greek to us

From New Scientist #2417, 18th October 2003 [link]

In the following addition sum digits have been consistently replaced by letters, with different letters used for different digits: Our alphabet has 26 characters whereas the Greek alphabet has only 24. Appropriately enough, ALPHABET is divisible by both 26 and 24. Also we can tell you that the third Greek letter GAMMA is divisible by 3.

Find DELEGATE.

[enigma1261]

Enigma 1306: Three all

From New Scientist #2464, 11th September 2004

I have in mind three numbers each of three digits (no leading zero) in each of which one digit is 3. Of the following statements about them, three are true and three are false.

(a) The number is a prime.
(b) The number is (appropriately) a cube.
(c) The middle digit is the average of the other two digits.
(d) The third digit differs from the second by 3.
(e) The number has as a factor a two-digit prime the difference of whose digits is 3, or whose sum is a cube.
(f) The number belongs (appropriately) to the set of triangular numbers 1, 3, 6, 10, 15, 21…

What is the sum of the three numbers?

[enigma1306]

Enigma 1319: Latin Christmas tree

From New Scientist #2477, 11th December 2004

My junior class has been studying Roman numerals. I asked them to replace each asterisk in the picture below with I, V or X so that each row formed a different valid Roman numeral, and either all were even or all were odd.

No letter was to appear in a row if it  did not also occur in all the rows below it. When we totted up all the numbers appearing in each design, we found that Alice’s sum was the same as Barbara’s, although the individual numbers in their designs were not all the same.

What was this score?

[enigma1319]

Enigma 1324: Rhombic squares

From New Scientist #2483, 22nd January 2005

Imagine you have two identical isosceles right-angled triangles. Lay one down so that one short side is horizontal and one is vertical. Lay the second one down so that a short side of the second is against a short side of the first, the two hypotenuses being parallel, so forming a rhombus, one example being as shown.

You are asked to place a digit at each of nine points, the four corners of the rhombus, the midpoint of each of the four sides, and the midpoint of the shorter diagonal such that eight numbers read horizontally and vertically from the top are all different perfect squares.

Alison and Bertha have each found a different solution. One of Alison’s two-digit squares is 81.

Which two did Bertha find?

[enigma1324]

Enigma 1337: A powerful square

From New Scientist #2496, 23rd April 2005

I have put one digit in each square of a 4×4 grid, so that I can read eight different four-figure numbers across the rows or down the columns, four of them being odd and four even. One of the across numbers is a cube and another is a fourth power, and the same is true of the numbers down. Two of the other numbers are squares.

Which two numbers in the grid are not perfect powers?

[enigma1337]

Enigma 1352: Old MacDonald

From New Scientist #2511, 6th August 2005

Older puzzlers know that 14 pounds make a stone and 8 stone make a hundredweight. Old MacDonald sold four varieties of potato, Primes, Queens, Roosters and Superb in 1-stone bags. One kind of bag, Chippers, contained an integral number of pounds of each variety, in order of increasing weight from Primes to Superbs; the other, Boilers, contained an integral number of pounds of each variety in order of increasing weight from Suberb to Primes. The number of pounds of each variety in each bag were different.

He also sold 1-hundredweight sacks of Commercial, which consisted of bags of Boilers and Chippers mixed. It so happened that in Commercial, half the weight of Superb came from Boiler bags and half from Chippers, and the number of pounds of Queens equalled the number of pounds of Roosters.

How many pounds of Primes were in 1 hundredweight of Commercial?

[enigma1352]

Enigma 1358: Five fives

From New Scientist #2517, 17th September 2005

You might think there is something wrong with the addition sum shown below, but in fact each of the five numbers shown in the sum is in a different base.

All five numbers are even and the given total at the bottom is less than 100,000.

What is that total?

[enigma1358]

Enigma 1364: Four all

From New Scientist #2523, 29th October 2005

I have in mind four numbers, each of four different digits one of which is 4. For each of them, four of the statements below are true and four are false.

(1) The number is a fourth power.
(2) The number is divisible by four.
(3) The number consists of a two-digit square followed by a smaller two-digit square.
(4) The product of the four digits exceeds the fourth power of 2.
(5) The number does not have two or more different prime factors.
(6) Each of the four digits is a perfect square.
(7) The digits form an arithmetic progression.
(8) The sum of the digits is prime, or else the sum of the two digits of that sum is 4.

What are my four numbers?

This is a similar type of problem to Enigma 1775.

[enigma1364]

Enigma 1367: Roman ladder

From New Scientist #2526, 19th November 2005

Your task this week is to put one of the letters I, V or X into each of the 21 squares in the ladder so that each row and column is a valid Roman numeral. All the numbers in the rows except one must be odd, and in only one case is a number in a row the same as a number in a column. If a letter appears in one row, it must appear in all rows above it.

What, from top to bottom, are the six numbers in the rows, expressed in the usual decimal form?

[enigma1367]

Enigma 1775: Third symphony

From New Scientist #2943, 16th November 2013 [link] [link]

I have in mind three numbers. Each is a multiple of 3 and consists of three different non-zero digits, just one of which is 3. For each of these numbers individually, three of the following six statements are true and three are false.

(1) It is the product of three different numbers, each a prime.
(2) It is a triangular number, that is of the series 1, 3, 6, 10, 15 …
(3) It is a cube, or a cube plus 3.
(4) It may be written as a single-digit prime followed by a two-digit prime.
(5) The 3 is the first or last digit.
(6) It may be written as a two-digit prime followed by a single-digit prime.

What are my three numbers?

[enigma1775]

Enigma 1374: Dominoes to the power of two

From New Scientist #2534, 14th January 2006

Using a standard set of dominoes, I laid out a row end-to-end such that (in typical domino style) touching ends were the same (e.g. 6-3, 3-5, 5-5, 5-1 etc). I started with 0-0 and then added dominoes so that the total number of pips in the row equalled 1, then 4, then 9, working my way though each perfect square and continuing as far as possible. My row could not have been shorter at any stage.

Having reached the highest square possible I was left with five dominoes, four of which were doubles.

What were those five dominoes?

[enigma1374]

Enigma 1378: Exaudi Deus

From New Scientist #2538, 11th February 2006

In our church we use a hymnbook with fewer than a thousand hymns. On Sunday we sing an opening hymn, a psalm, and three more hymns, the numbers being displayed in that order on the board.

Recently, as I sat at the organ I noticed that the five numbers were all different perfect squares in increasing order, and that each digit which did occur, occurred exactly three times.

What was the number of the psalm?

[enigma1378]

Enigma 1384: Power grid

From New Scientist #2544, 25th March 2006

I have drawn a grid of five rows of nine boxes and have placed in each box a digit so that each digit occurs its own number of times (e.g. 4 occurs four times). The digits in one row form a number which is a fourth power, and another row contains only two different digits and is divisible by 33. Two other rows between them contain all the digits and are both fifth powers.

In ascending order, what are the nine digits occupying the remaining row?

[enigma1384]

Enigma 1770: Power point 2

From New Scientist #2938, 12th October 2013 [link] [link]

I have written a list of five different three-figure numbers, each of which is a power of a single digit. The first number is odd and thereafter each number has the same hundreds digit or the same tens digit or the same units digit as its predecessor.

What (in order) are the five numbers?

This puzzle (apart from a comma) is exactly the same as Enigma 1757.

New Scientist has stated that the puzzle was republished in error.

[enigma1770]

Enigma 1395: A matter of fives

From New Scientist #2555, 10th June 2006

I have in mind five five-digit numbers. For each of them individually, five of the following 10 statements are true:

(a) It contains two zeroes;
(b) It is a perfect cube;
(c) It is the product of five successive numbers;
(d) It is a perfect square;
(e) The central digit is a 5;
(f) The sum of the digits is a square or cube, or has both its digits the same;
(g) It is not prime but has only one prime factor;
(h) It is a power greater than the third;
(i) It is a palindromic square of a palindrome;
(j) The first and last digits put together in that order form a perfect square.

In ascending order, what are my five numbers?

[enigma1395]

Enigma 1764: Secret passages

From New Scientist #2932, 31st August 2013 [link] [link]

Kathryn and her school friends have been using a Lorenz-type code to pass covert messages to each other. Each letter is expressed as a five-digit binary number such that A = 1 = 00001, M = 13 = 01101 and so on, but other symbols are represented by 00000 and by 11011 upwards. A fixed letter, say M, is chosen as a “coder”, known only to the sender and receiver. To transmit a letter, say D, it is added to the coder by the “exclusive-NOR” rule:

1 + 1 = 1,
1 + 0 = 0,
0 + 1 = 0,
0 + 0 = 1.

So, for example, D + M = 00100 + 01101 = 10110 = V. When the sent letter V is added by the recipient to the coder M, the original letter reappears: 10110 + 01101 = 00100.

She has sent her name to her friends as seven letters. KATHRYN and its coded version together consist of 14 different letters, so what was the coded version?

[enigma1764]